Beyond Sperner's lemma

TL;DR

This paper revisits Scarf's combinatorial proof of Brouwer's fixed point theorem, emphasizing the geometric structure behind the abstract combinatorial arguments and extending the proof to a more general setting.

Contribution

It provides a new proof of Scarf's Sperner's lemma analogue in an abstract setting, restoring geometric intuition and deriving Brouwer's theorem from it.

Findings

Restores geometric aspects in the combinatorial proof

Extends Scarf's lemma to abstract linear orders

Derives Brouwer's fixed point theorem from the analogue

Abstract

In 1967 Herbert Scarf suggested a new proof of Brouwer's fixed point theorem based on a combinatorial analogue of Sperner's lemma. Scarf presented his arguments in very geometric language, even purely combinatorial ones. Recently H. Petri and M. Voorneveld published an almost geometry-free version of Scarf's proof. Their version eliminated even only implicitly geometric aspects of Scarf's proof, namely, the structure of an abstract simplicial complex behind the combinatorial arguments. The present paper is devoted to a proof of Scarf's analogue of Sperner's lemma in the abstract setting of a collection of linear orders on a finite set. This proof partially follows the proof by Petri and Voorneveld, but restores the implicit geometry to its rightful place. We also deduce Brouwer's fixed point theorem from this analogue and discuss various versions of Scarf's proof.

Equations

$$\text{mbox}thereisnoelementy\in T\text{mbox}suchthat{\min }_i\sigma {<}_{i}y\text{mbox}foralli\in C.$$

$${\min }_a\tau ={\min }_b\tau$$

$$\quad\mathbb{M}_{\hskip 0.70004pti}\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt\bigl{\{}\hskip 3.99994pty\hskip 1.99997pt\in\hskip 1.99997ptT\hskip 3.99994pt\bigl{|}\bigr{.}\hskip 3.99994pt\min\nolimits_{\hskip 0.70004ptk}\hskip 1.00006pt\tau\hskip 3.99994pt<_{k}\hskip 3.99994pty\hskip 3.99994pt\mbox{for\hskip 1.00006pt\ all}\hskip 3.99994ptk\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 1.99997pt-\hskip 1.99997pti\hskip 3.99994pt\bigr{\}}\hskip 1.00006pt.$$

$${\min }_i(\tau +x)=x\text{mbox}ifx{<}_i{\min }_i\tau ,\text{mbox}and$$

$${\min }_i(\tau +x)={\min }_i\tau \text{mbox}if{\min }_i\tau {<}_{i}x.$$

$$\quad\bigl{\{}\hskip 3.00003pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\tau\hskip 3.99994pt|\hskip 3.99994pti\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 3.00003pt\bigr{\}}\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.00006pt\tau\hskip 15.00002pt\mbox{and\hskip 1.99997pt\ that}\hskip 15.00002pt$$

$$\quad\bigl{\{}\hskip 3.99994pt\min\nolimits_{\hskip 0.35002pti}\hskip 3.00003pt(\hskip 1.00006pt\tau\hskip 1.99997pt+\hskip 1.99997ptx\hskip 1.00006pt)\hskip 3.99994pt\bigl{|}\bigr{.}\hskip 3.99994pti\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 3.99994pt\bigr{\}}\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt\tau\hskip 3.99994pt+\hskip 3.99994ptx$$

$${\min }_i(\tau +x)={\min }_i\tau \text{mbox}foralli\in D-a\text{mbox}and$$

$${\min }_a(\tau +x)=x.$$

$${\min }_i\tau {<}_{i}x\text{mbox}foralli\in D-a.$$

$$|\sigma |⩽|C|⩽|c(\sigma )|+1⩽|\sigma |+1.$$

$$|\sigma |⩾|c(\sigma )|⩾|c(\sigma )\cap C|=|C|-1=|\sigma |$$

$$C\setminus c(\sigma -x)=(C\setminus c(\sigma ))+c(x)$$

$$C\setminus c(\sigma -x)=C\setminus c(\sigma )$$

$$(C+i)\setminus c(\sigma )=(C\setminus c(\sigma ))+i$$

$$(C+i)\setminus c(\sigma )=C\setminus c(\sigma )$$

$$(\tau ,D)\text{mbox}isadoorof(\sigma ,C)$$

$$x_i<y_i\text{mbox}impliesx{<}_{i}y$$

$$|x_i-y_i|<2(n+1)/l$$

$$0⩽x_i-m_i<|C|/l$$

$$1-\sum _{k\in C}{m}_k<|C|/l.$$

$$\quad\sum_{k\hskip 1.39998pt\in\hskip 1.39998ptC\vphantom{K^{K}}}\hskip 1.99997pt\bigl{(}\hskip 1.99997ptm_{\hskip 0.70004ptk}\hskip 1.99997pt+\hskip 1.99997pt(\hskip 1.00006pt1\hskip 0.50003pt/\hskip 0.50003ptl\hskip 1.99997pt)\hskip 1.99997pt\bigr{)}\hskip 3.99994pt\hskip 1.99997pt=\hskip 3.99994pt\hskip 3.99994pt\left(\hskip 3.99994pt\sum_{k\hskip 1.39998pt\in\hskip 1.39998ptC\vphantom{K^{K}}}\hskip 1.99997ptm_{\hskip 0.70004ptk}\hskip 3.99994pt\right)\hskip 3.99994pt+\hskip 3.99994pt\hskip 1.00006pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 0.50003pt\left/\hskip 1.00006ptl\right.\hskip 3.99994pt\hskip 3.99994pt\leqslant\hskip 3.99994pt\hskip 3.99994pt1\hskip 1.99997pt.$$

$$x_i-m_i⩽\sum _{k\in C}\left(x_k-m_k\right)⩽\sum _{k\in I}{x}_k-\sum _{k\in C}{m}_k=1-\sum _{k\in C}{m}_k.$$

$$x_i-m_i=x_i⩽\sum _{k\notin C}{x}_k=1-\sum _{k\in C}{x}_k⩽1-\sum _{k\in C}{m}_k.$$

$$x=(x_0,x_1,\dots ,x_n)\in T_l\text{mbox}andy=(y_0,y_1,\dots ,y_n)=f(x),$$

$$w=(w_0,w_1,\dots ,w_n)=f(z).$$

$${\sum }_{i\in C}{z}_i=1.$$

$${\sum }_{i\in I}{w}_i=1.$$

$$s(i)=(M_i,\dots ,M_i,0,M_i,\dots ,M_i)\in {\mathbf{R}}^{n+1},$$

Full text

††footnotetext: ©   Nikolai  V.  Ivanov,   2017,  2022  (revised  version).   The author  is  grateful  to  V. I.  Danilov  for attracting  his attention  to  the works of  Scarf  and  very  stimulating  correspondence.

Beyond  Sperner’s  lemma

Nikolai  V.  Ivanov

Contents

Preface 1  

1****1.   Scarf’s  combinatorial  theorem  in  an abstract  setting 3  

1****2.   Brouwer’s  fixed  point  theorem 9  

1****3.   Scarf’s  proof  and  its  versions 11  

References   14

Preface

In  1967  H.  Scarf  [$S\mathrm{c}_{\hskip 0.70004pt2}$]  suggested a new  proof  of  Brouwer’s  fixed  point  theorem.   In outline,   his proof  is  similar  to  the well  known  proof  based on  Sperner’s  lemma and  Knaster–Kuratowski–Mazurkiewicz  [KKM]  argument .   But  instead of  triangulations  Scarf  uses  (sufficiently  dense)  finite subsets of  a simplex .   This  forces a modification of  the  KKM  argument :   the inequalities in  the coloring  rule must  be reversed.   Also,   instead of  using  the fundamental  fact  that  an odd  number cannot  be equal  zero,   Scarf  use a path-following  algorithm  motivated  by  the  game  theory and  the  linear programming.   From  Scarf’s  point  of  view,   the algorithmic nature of  his proof  was especially  important.   But,   as explained  in  [$I_{\hskip 0.70004pt1}$],   the proof  based on  Sperner’s  lemma can be easily  turned  to a path-following algorithm.   Moreover,   path-following  arguments we used already  in  1929  by  Hurewicz  [H].

The most  striking  feature of  Scarf’s  proof  is  a  combinatorial  theorem,   namely,   an analogue of  Sperner’s  lemma  dealing  with  finite subsets of  a simplex  instead of  triangulations.   See  [$S\mathrm{c}_{\hskip 0.70004pt2}$],   Theorem  1.   The origins of  this  theorem and  its  proof  also belong  to  the  linear  programming  and  the game  theory .   Actually ,   Scarf’s  proof  of  Brouwer’s  theorem  is  a  byproduct  of  Scarf’s  fundamental  paper  [$S\mathrm{c}_{\hskip 0.70004pt1}$]  in  game  theory .   A  topological  interpretation of  Scarf’s  combinatorial  theorem was suggested  by  H.  Kuhn.   See  [Ku],   Scarf’s  Theorem.

Scarf’s  combinatorial  theorem  is  a  truly  combinatorial  result ,   but  the  style of  his papers  is  very  geometric,  and  he always puts forward  the geometric underpinnings of  combinatorial  arguments.   For example,   Scarf  orders subsets of  a simplex  by  the values of  one of  barycentric coordinates,   but  instead of  taking  the minimal  element  of  a  set  he  moves a face of  a simplex  parallely  to  itself  (see  [$S\mathrm{c}_{\hskip 0.70004pt2}$],   the proof  of  Lemma  1,  for example).

Recently  H.  Petri  and  M.  Voorneveld  [PV]  published  an almost  geometry-free  version of  Scarf’s  proof  of  Brouwer’s  theorem.   Eliminating  geometry  is  an explicitly  stated  goal  of  [PV].   See  [PV],   Concluding  remarks.   This  is  achieved  not  without  a cost ,   Petri–Voorneveld  version  of  Scarf’s  proof  is  fairly  tricky .   A minimal  amount  of  geometry  is  inevitable since  Brouwer’s  theorem  is  about  a simplex .   But  the geometry  of  simplices  is  eliminated  by  dealing  with  a  family  of  arbitrary  linear  orders on a  finite set  instead of  the linear orders induced  by  barycentric coordinates on a subset  of  a simplex .   Scarf  himself  had applied  his  methods  to other orders and  was aware  that  they  work for arbitrary  orders.   See  [$S\mathrm{c}_{\hskip 0.70004pt3}$],   Chapter  6  and,  especially ,  the discussion at  the bottom of  p.  146.   Rewriting  Scarf’s  arguments  in  such an abstract  setting  is  only  natural  and  makes  their combinatorial  nature  transparent .

But  Petri  and  Voorneveld  went  further and eliminated  also only  implicitly  geometric aspects of  the proof .   Like  Sperner  [S],   Petri  and  Voorneveld  [PV]  start  with a coloring  of  a finite set .   In  Sperner’s  context  this finite set  is  the set  of  vertices of  a  triangulation,   and  this  triangulation  is  present  before any  coloring  is  introduced  and  even  before  the proof  begins.   A similar  structure on an arbitrary  finite set  of  points  in a  simplex  (or  a  finite set  with a  family  of  linear orders)  was discovered  by  Scarf  [$S\mathrm{c}_{\hskip 0.70004pt1}$],   [$S\mathrm{c}_{\hskip 0.70004pt2}$].   It  is  the collection of  so-called  primitive  sets,   and  it  was  interpreted  by  Kuhn  [Ku]  as  the structure of  a simplicial  complex .   See  Section  3.   Scarf’s  proof  and  its  versions  below.   Petri  and  Voorneveld  use only  (a version of )  primitive sets closely  related  to  the coloring  in  question,  and  this renders  the  Scarf–Kuhn  structure  invisible.

The present  paper  originated  in  a  text  written  simply  to verify  Petri–Voorneveld  proof .   After  the verification,   the proof  was rearranged  in order  to stress similarities with  Sperner’s  proof  of  his  lemma.   The rearranged  proof  turned out  to be equally  well  suited  for comparing  with  Scarf’s   one.   After  such a comparison  it  became clear  that  the  Scarf–Kuhn  structure should  be separated  from colorings.

The main  goal  of  this  paper  is  to present  the resulting  proof  of  Scarf’s  combinatorial  theorem  in  the setting of  a  family  of  orders on a  finite set .   This  is  done in  Section  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.   Section  2.   Brouwer’s  fixed  point  theorem  is  devoted  to a deduction of  Brouwer’s  fixed  point  theorem  from  Scarf’s  theorem.   This deduction  follows  Petri  and  Voorneveld  [PV]  version.   Since  the geometry  is  avoided,   their  version  is  a  little more complicated  than  Scarf’s  one.   Finally ,   Section  3.   Scarf’s  proof  and  its  versions  explains how  the proofs  in  [PV]  and  in  the present  paper are related  to  Scarf’s  proofs.   In a  related  paper  [$I_{\hskip 0.70004pt2}$]  Scarf’s  results  are  discussed  from  the point  of  view of  the combinatorial  topology ,   and,   in  particular ,   some ideas  of  Kuhn  [Ku]  are refined.   See  [$I_{\hskip 0.70004pt2}$],   Sections  1 – 3.

1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting

Notations.   Let  $A$  be set .   For  $a\hskip 1.99997pt\in\hskip 1.99997ptA$  we denote by  $A\hskip 1.99997pt-\hskip 1.99997pta$  the set  $A\hskip 1.99997pt\smallsetminus\hskip 1.99997pt\{\hskip 1.99997pta\hskip 1.99997pt\}$,   and  for  $b\hskip 1.99997pt\not\in A$  wel denote by  $A\hskip 1.99997pt+\hskip 1.99997ptb$  the set  $A\hskip 1.99997pt\cup\hskip 1.99997pt\{\hskip 1.99997ptb\hskip 1.99997pt\}$.   The set  $A\hskip 1.99997pt-\hskip 1.99997pta$  is defined only  if  $a\hskip 1.99997pt\in\hskip 1.99997ptA$,   and  $A\hskip 1.99997pt+\hskip 1.99997ptb$  is defined only  if  $b\hskip 1.99997pt\not\in\hskip 1.99997ptA$.   By  $|\hskip 1.99997ptA\hskip 1.99997pt|$ we denote the number of  elements of  $A$.

Linear orders and  dominant sets.   Let $T$ be a finite set .   Suppose that a family of  linear orders $<_{\hskip 0.70004pti}$ on  $T$,   labeled  by elements $i$ of  a  finite set $I$,  is given.   For a non-empty  subset  $\sigma\hskip 1.99997pt\subset\hskip 1.99997ptT$  let  $\min_{\hskip 0.70004pti}\hskip 1.99997pt\sigma$  be the minimal element of $\sigma$ with respect to the order  $<_{\hskip 0.70004pti}$.   A subset  $\sigma\hskip 1.99997pt\subset\hskip 1.99997ptT$  is  said  to  be  dominant  with  respect  to  a non-empty subset  $C$  of  $I$  if

[TABLE]

It is convenient  to agree that  $\varnothing\hskip 1.99997pt\subset\hskip 1.99997ptT$  is dominant with respect to every non-empty  $C\hskip 1.99997pt\subset\hskip 1.99997ptI$.   The basic properties of  dominant  sets are summarized  in  the following  lemma.

1. Lemma.   *If  $\sigma\hskip 1.99997pt\subset\hskip 1.99997ptT$  is  dominant  with  respect  to  $C\hskip 1.99997pt\subset\hskip 3.99994ptI$,   then  \displaystyle\sigma\hskip 3.99994pt=\hskip 3.99994pt\bigl{\{}\hskip 1.99997pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.00006pt\sigma\hskip 3.99994pt\bigl{|}\bigr{.}\hskip 3.99994pti\hskip 1.99997pt\in\hskip 1.99997ptC\hskip 1.99997pt\bigr{\}}  and  hence  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|$.   If   $\tau\hskip 3.99994pt\subset\hskip 3.99994pt\sigma$,   then  $\tau$  is  dominant  with  respect  to  $C$,   and  if   $C\hskip 3.99994pt\subset\hskip 3.99994ptD$,   then  $\sigma$  is  dominant  with  respect  to  $D$.   *

Proof**.**   Clearly ,  $\sigma$ contains all minima  $\min_{\hskip 0.35002pti}\hskip 1.00006pt\sigma$.   Suppose that  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  is different from all  $\min_{\hskip 0.35002pti}\hskip 1.00006pt\sigma$  with  $i\hskip 1.99997pt\in\hskip 1.99997ptC$.   Then  $\min_{\hskip 0.35002pti}\hskip 1.00006pt\sigma\hskip 1.99997pt<_{\hskip 0.35002pti}\hskip 1.99997ptx$  for all  $i\hskip 1.99997pt\in\hskip 1.99997ptC$,   contrary to the assumption.   This proves  the  first  statement  of  the lemma.   The second  statement  follows from  the fact  that  $\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\sigma\hskip 3.99994pt\leqslant_{\hskip 0.70004pti}\hskip 3.99994pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\tau$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$,   and  the  third  statement  is  obvious.    $\blacksquare$

Cells,   rooms,   and  doors.   A  cell   is  defined as  a  pair $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$ of  subsets  $\sigma\hskip 1.99997pt\subset\hskip 1.99997ptT$  and  $C\hskip 1.99997pt\subset\hskip 3.00003ptI$  such  that  $C$ is  non-empty  and $\sigma$ is  dominant with respect  to  $C$.   If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a cell ,   then  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt\leqslant\hskip 1.99997pt|\hskip 1.99997ptC\hskip 1.99997pt|$  by  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.   A cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  called  a  room   if  $|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$,   and  a  door   if  $|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1$.

A  pair  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  of  subsets  $\tau\hskip 1.99997pt\subset\hskip 1.99997ptT$  and  $D\hskip 1.99997pt\subset\hskip 3.00003ptI$  is  said  to  be a  door  of  a  cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  if  either  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  for  some  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$,    or  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)$  for some  $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   By  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  every  such  pair  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  a  cell .   It  follows  that  every  room  has  $|\hskip 1.99997ptI\hskip 1.99997pt|$  doors.

Equivalently ,  a  pair  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  door  of  a  cell  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  if  either  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997pty\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  for some  $y\hskip 1.99997pt\in\hskip 3.00003ptT\hskip 1.99997pt\smallsetminus\hskip 1.99997pt\sigma$,   or  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.99997pt-\hskip 1.99997ptj\hskip 1.49994pt)$  for some  $j\hskip 1.99997pt\in\hskip 3.00003ptD$.

Clearly ,   if  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  door  of  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$,   then  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt|\hskip 1.99997pt\tau\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1$.   It  follows  that  if  a  cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a  room,   then  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  not  a  door of  any  cell .   Similarly ,   if  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  door of  a cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$,   then  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a  room .   A door  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  called  an  outside  door   if  $\tau\hskip 3.99994pt=\hskip 3.99994pt\varnothing$  and  an  internal  door  otherwise.

One can  imagine  that  $T$  together  with  the rooms and doors  is  some sort  of  a  building .   As we will  see,   an outside door  is  a  door of  exactly  one room,   and  an  internal  door  is  a  common  door  of  exactly  two rooms.   These properties are  the main  properties of  this  building .

2. Lemma.     *Suppose  that  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.   Then  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  an outside door  and  is  a door  of  exactly  one room.   Every  outside door  has  this  form.   *

Proof**.**   Since $\varnothing$ is  dominant  with  respect  to every  non-empty  subset  of  $I$,   the pair  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a cell .   Since  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$,   this cell  is  a  door  and  hence an outside door .   Also,  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$  implies  that  $D\hskip 1.99997pt=\hskip 1.99997pt\{\hskip 1.49994pti\hskip 1.99997pt\}$  for some  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.   If  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  is  a  door of  a cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)$,   then  either  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.99997pt+\hskip 1.99997ptj\hskip 1.49994pt)$  for some  $j\hskip 1.99997pt\not\in\hskip 1.99997pt\{\hskip 1.49994pti\hskip 1.99997pt\}$,   i.e.  for some  $j\hskip 3.99994pt\neq\hskip 3.99994pti$,   or  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.49994pt\{\hskip 1.49994ptx\hskip 1.99997pt\}\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  for some  $x\hskip 1.99997pt\in\hskip 1.99997ptT$.   In  the first  case  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt0$  and  $|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt2$  and  hence  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)$  is  not  a  room.   In  the second case $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)$ is  a  cell  if  and  only  if  $\{\hskip 1.49994ptx\hskip 1.99997pt\}$ is dominant with respect to  $\{\hskip 1.49994pti\hskip 1.99997pt\}$,   i.e.  if  and  only  if  $x$  is  the maximal  element  of  $T$  with  respect  to  $<_{\hskip 0.70004pti}$.   It  follows  that  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  is  a  door  of  exactly  one room.   Clearly ,  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  door only  if  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.    $\blacksquare$

3. Lemma.     *An  internal  door  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  door  of  exactly  two  rooms.   *

Proof**.**   Since  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  an  internal  door ,   $|\hskip 1.99997pt\tau\hskip 1.99997pt|\hskip 1.99997pt\geqslant\hskip 1.99997pt1$  and  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\tau\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1\hskip 3.99994pt\geqslant\hskip 3.99994pt2$.   Suppose  that  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$ is  a  door of  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$.   Then  either  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptx\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  for some  $x\hskip 1.99997pt\not\in\hskip 3.00003pt\tau$,   or  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.99997pt-\hskip 1.99997pti\hskip 1.49994pt)$  for some  $i\hskip 1.99997pt\in\hskip 3.00003ptD$.   Since  $|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 3.99994pt\geqslant\hskip 3.99994pt2$,   the set  $C$  is  non-empty .   Therefore,   if  $\sigma$ is  dominant  with  respect  to $C$,   then  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a  cell  and  hence  is  a  room.   Let  us  find  out  when $\tau\hskip 1.99997pt+\hskip 1.99997ptx$  is  dominant  with  respect  to  $D$  and  $\tau$  is  dominant  with  respect  to  $D\hskip 1.99997pt-\hskip 1.99997pti$. Since  $|\hskip 1.99997pt\tau\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1$,   Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  implies that there is a unique pair  $\{\hskip 1.00006pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}\hskip 1.99997pt\subset\hskip 3.00003ptD$  such  that

[TABLE]

and  $a\hskip 1.99997pt\neq\hskip 1.99997ptb$.    For   $i\hskip 3.99994pt=\hskip 3.99994pta$  or  $b$   let

[TABLE]

When  $\mathbb{M}_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$,   we will  denote  by $m_{\hskip 0.70004pti}$ the maximal  element  of  $\mathbb{M}_{\hskip 0.70004pti}$  with respect  to  $<_{\hskip 0.70004pti}$.   Since $\tau$ is  dominant  with  respect  to  $D$,   the  intersection  $\mathbb{M}_{\hskip 0.70004pta}\hskip 1.99997pt\cap\hskip 3.00003pt\mathbb{M}_{\hskip 0.70004ptb}$  is  empty .   Therefore,   if  both  $\mathbb{M}_{\hskip 0.70004pta}$  and  $\mathbb{M}_{\hskip 0.70004ptb}$  are non-empty ,   then  $m_{\hskip 0.70004pta}\hskip 3.99994pt\neq\hskip 3.99994ptm_{\hskip 0.70004ptb}$  and  hence  the  pairs  $(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptm_{\hskip 0.70004pta}\hskip 1.00006pt,\hskip 3.00003ptD\hskip 1.49994pt)$  and  $(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptm_{\hskip 0.70004ptb}\hskip 1.00006pt,\hskip 3.00003ptD\hskip 1.49994pt)$  are different .   This reduces  the  lemma  to  the following  two sublemmas.

3.1. Sublemma.   The set  $\tau$  is  dominant with respect  to  $D\hskip 1.99997pt-\hskip 1.99997pti$  if  and  only  if  $i\hskip 1.99997pt\in\hskip 1.99997pt\{\hskip 1.00006pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}$  and  $\mathbb{M}_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt\varnothing$.

Proof.     If  $i\hskip 3.99994pt\neq\hskip 3.99994pta\hskip 0.50003pt,\hskip 3.00003ptb$,   then  the set  \displaystyle\bigl{\{}\hskip 3.00003pt\min\nolimits_{\hskip 0.70004ptk}\hskip 1.00006pt\tau\hskip 3.99994pt|\hskip 3.99994ptk\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 1.99997pt-\hskip 1.99997pti\hskip 3.00003pt\bigr{\}}  has  $\leqslant\hskip 1.99997pt|\hskip 1.99997ptD\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt2\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\tau\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$   elements  and  hence  $\tau$  is  not  dominant  with  respect  to  $D\hskip 1.99997pt-\hskip 1.99997pti$  by  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.   If  $i\hskip 3.99994pt=\hskip 3.99994pta$  or  $b$,   then  $\tau$  is  dominant  with respect  to  $D\hskip 1.99997pt-\hskip 1.99997pti$  if  and  only  if  $\mathbb{M}_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt\varnothing$.    $\square$

3.2. Sublemma.   *$\tau\hskip 1.99997pt+\hskip 1.99997ptx$  is  dominant  with  respect  to  $D$  if  and  only  if  $x\hskip 3.99994pt=\hskip 3.99994ptm_{\hskip 0.70004pti}$  for  some  $i\hskip 1.99997pt\in\hskip 3.00003pt\{\hskip 1.00006pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}$  such  that   $\mathbb{M}_{\hskip 0.70004pti}\hskip 3.99994pt\neq\hskip 3.99994pt\varnothing$.   *

Proof.   To begin with,   let us observe that

[TABLE]

[TABLE]

In particular ,   $\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\tau$   or   $x$   for every   $i\hskip 1.99997pt\in\hskip 1.99997ptD$.   Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  implies  that

[TABLE]

[TABLE]

if  $\tau\hskip 1.99997pt+\hskip 1.99997ptx$  is  dominant  with  respect  to $D$.   This may happen only  if  $\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\tau$   for all  $i\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 1.99997pt\smallsetminus\hskip 1.99997pt\{\hskip 1.99997pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}$  and  for $i$ equal  to one of  the elements of  the pair  $\{\hskip 1.00006pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}$,   and  if  $\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt(\hskip 1.49994pt\tau\hskip 1.99997pt+\hskip 1.99997ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptx$  for $i$ equal  to  the  other element  of  $\{\hskip 1.00006pta\hskip 0.50003pt,\hskip 3.00003ptb\hskip 1.99997pt\}$.

Therefore,   if  $\tau\hskip 1.99997pt+\hskip 1.99997ptx$  is  dominant  with  respect  to $D$,   we may  assume  that

[TABLE]

[TABLE]

By  (1)  and  (2)  in  this case   $\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\tau\hskip 3.99994pt<_{\hskip 0.70004pti}\hskip 3.99994ptx$   for all  $i\hskip 1.99997pt\in\hskip 1.99997ptD\hskip 1.99997pt-\hskip 1.99997pta$  and  $x\hskip 3.99994pt<_{\hskip 0.70004pta}\hskip 3.99994pt\min\nolimits_{\hskip 0.70004pta}\hskip 1.99997pt\tau$.   It  follows  that  $x\hskip 1.99997pt\in\hskip 1.99997pt\mathbb{M}_{\hskip 0.70004pta}$  and  that  $\tau\hskip 1.99997pt+\hskip 1.99997ptx$  can  be dominant with  respect  to  $D$  only  if  $x$  is  the maximal  element  of  $\mathbb{M}_{\hskip 0.70004pta}$  with respect  to  $<_{\hskip 0.70004pta}$,   i.e.  only  if  $x\hskip 3.99994pt=\hskip 3.99994ptm_{\hskip 0.70004pta}$.

Conversely ,   if ,   say ,   $\mathbb{M}_{\hskip 0.70004pta}\hskip 1.99997pt\neq\hskip 1.99997pt\varnothing$  and  $x\hskip 1.99997pt\in\hskip 1.99997pt\mathbb{M}_{\hskip 0.70004pta}$,   then

[TABLE]

If  also  $\min\nolimits_{\hskip 0.70004pta}\hskip 1.99997pt\tau\hskip 3.99994pt<_{\hskip 0.70004pta}\hskip 3.99994ptx$,   then  $\tau$  is  not  dominant  with  respect  to  $D$,   contrary  to  the assumption.   Therefore  $x\hskip 3.99994pt<_{\hskip 0.70004pta}\hskip 3.99994pt\min\nolimits_{\hskip 0.35002pta}\hskip 1.99997pt\tau$.   By applying  (1)  and  (2)  again,   we see that  (3)  and  (4)  hold.   It  follows  that  if  $x\hskip 3.99994pt=\hskip 3.99994ptm_{\hskip 0.70004pta}$,   then  $\tau\hskip 1.99997pt+\hskip 1.99997ptx$  is  dominant  with  respect  to  $D$.    $\square$ $\blacksquare$

Colorings.   A  coloring  is  arbitrary  map  $T\hskip 1.99997pt\longrightarrow\hskip 1.99997ptI$.   Let  us  fix  a  coloring  $c$.   A  cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  called  colorful   if   $C\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)$.   A colorful  cell  is  automatically  a  room.   Scarf’s  combinatorial  theorem  asserts  that  for every  coloring  there exists a  colorful  room.   See  Theorem  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.

A cell  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  called  nearly  colorful   if   $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.   If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  nearly  colorful,   then  $C\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$  consists of  one element ,   called  the  type  of  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$.

4. Lemma.   *For every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  there  is  exactly  one outside door of  the  type  $i$.    Every  door of  a colorful  room  is  nearly  colorful  and  there  is  exactly  one door of  each  type among  them.   *

Proof**.**   The outside doors are  the cells of  the form  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$,   where  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.   Clearly ,   the door  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  is  nearly  colorful  and  its  type  is  $i$.   This proves  the first  statement .

Let  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  be a colorful  room.   If  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$,   then  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.49994pt(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.49994pt)\hskip 1.99997pt|\hskip 1.99997pt=\hskip 1.99997pt1$  and  hence  the door  $(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)$  is  nearly  colorful .   If  $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$,   then  $|\hskip 1.99997pt(\hskip 1.99997ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.99997pt)\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.49994pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt|\hskip 1.99997pt=\hskip 1.99997pt1$  and  hence  the door  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.99997pt+\hskip 1.99997ptx\hskip 1.49994pt)$  is  nearly  colorful .   Obviously ,   the  type of  $(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.49994pt)$  is  $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$,   and  the  type of  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)$  is $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   Since  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  colorful ,   the map  $c$ induces a  bijection  $\sigma\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptC$.   The  second  statement  follows.    $\blacksquare$

5. Lemma.   *Suppose  that  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$  is  a  nearly  colorful  door  of  a  room  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$.   Then  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  either  colorful ,   or  nearly  colorful  of  the  same  type  as  $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$.   *

Proof**.**   Clearly ,  $C\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt\subset\hskip 3.99994ptD\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\tau\hskip 1.49994pt)$.   Together  with  $|\hskip 1.99997ptD\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\tau\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$  this  implies  that  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt1$.   Since  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|$,    if  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt0$,   then  $C\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$  and  hence $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  colorful .   If  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$,   then  $C\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptD\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\tau\hskip 1.49994pt)$  and  hence  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  nearly  colorful  and  has  the same  type as $(\hskip 1.49994pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.49994pt)$.    $\blacksquare$

6. Lemma.   *If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$ is  a  nearly  colorful  room,   then  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|$  is  equal  either  to  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|$  or  to  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$.   If   $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a  nearly  colorful  door ,   then  $c\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 3.99994pt\subset\hskip 3.99994ptC$.   *

Proof**.**   If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$ is  a  nearly  colorful  room,   then  $|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1$  and  hence

[TABLE]

It  follows  that  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$  or  $|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$.   If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$ is  a  nearly  colorful  door ,   then

[TABLE]

and  hence  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt\cap\hskip 1.99997ptC\hskip 1.99997pt|$  and  $c\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt\cap\hskip 1.99997ptC$.   It  follows  that  $c\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt\subset\hskip 1.99997ptC$.    $\blacksquare$

7. Lemma.   If  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  a  nearly  colorful  room,   then  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  has  two  nearly  colorful  doors.

Proof**.**   Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  implies  that  either  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$  or  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$.

Suppose  first  that  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$.   Then  $c\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 3.99994pt\subset\hskip 3.99994ptC$.   Therefore,   if  also  $i\hskip 1.99997pt\not\in\hskip 3.00003ptC$,   then  $|\hskip 1.99997pt(\hskip 1.49994ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt2$  and  hence  $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)$  is  not  nearly  colorful .   Since $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|\hskip 1.99997pt-\hskip 1.99997pt1$,   there are different  elements  $x\hskip 0.50003pt,\hskip 3.00003pty\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  such  that  $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.00006pty\hskip 1.49994pt)$.   Moreover ,   the coloring  $c$  is  injective on  both  $\sigma\hskip 1.99997pt-\hskip 1.99997ptx$  and  $\sigma\hskip 1.99997pt-\hskip 1.99997pty$.   If  $z\hskip 1.99997pt\in\hskip 3.00003ptC$  and  $z\hskip 3.99994pt\neq\hskip 3.99994ptx\hskip 0.50003pt,\hskip 3.00003pty$,   then  $c\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.49994pt)$  is  properly  contained  in  $c\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$.   Since  $c\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.49994pt)$  is  properly  contained  in  $C$,   it  follows  that  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt2$  and  hence  $(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.00006pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  not  nearly  colorful .   On  the other  hand,   if  $z\hskip 3.99994pt=\hskip 3.99994ptx$  or  $y$,   then  $c\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)$  and  hence  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.   In  this case  $(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptz\hskip 1.00006pt,\hskip 1.99997ptC\hskip 1.49994pt)$  is  nearly  colorful .   Therefore  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  has  two  nearly  colorful  doors.

Suppose  that  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997pt\sigma\hskip 1.99997pt|$.   Then $c$ is  injective on $\sigma$.   Since $(\hskip 1.49994pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$ is  nearly  colorful ,   this  implies  that  $|\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma\hskip 1.00006pt)\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$ and  hence  there  is  a  unique  $y\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  such  that  $c\hskip 1.00006pt(\hskip 1.00006pty\hskip 1.49994pt)\hskip 1.99997pt\not\in\hskip 1.99997ptC$.   If  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$,   then  $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt\not\in\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.49994pt)$  because $c$ is  injective on $\sigma$.   If  also $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 3.00003ptC$,   then

[TABLE]

and  hence  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt2$.   Therefore  in  this case  $(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.00006pt)$  is  not  nearly  colorful .

On  the other  hand,   if  $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt\not\in\hskip 3.00003ptC$,   i.e.  if  $x\hskip 3.99994pt=\hskip 3.99994pty$,   then

[TABLE]

and  hence  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.   Therefore  $(\hskip 1.00006pt\sigma\hskip 1.99997pt-\hskip 1.99997ptx\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.00006pt)$ is  nearly  colorful  if  and  only  if  $x\hskip 3.99994pt=\hskip 3.99994pty$.    If  $i\hskip 1.99997pt\in\hskip 1.99997ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$  and  $i\hskip 1.99997pt\not\in\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)$,   then

[TABLE]

and  hence  $|\hskip 1.99997pt(\hskip 1.00006ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.99997pt=\hskip 1.99997pt2$.   Therefore  in  this case  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)$  is  not  nearly  colorful .   On  the other  hand,   if  $i\hskip 1.99997pt\in\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)$,   i.e.  if  $i\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994pty\hskip 1.49994pt)$,   then

[TABLE]

and  hence  $|\hskip 1.99997pt(\hskip 1.99997ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.99997pt)\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994pt1$.   Therefore  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 3.00003ptC\hskip 1.99997pt+\hskip 1.99997pti\hskip 1.49994pt)$ is  nearly  colorful  if  and  only  if  $i\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.49994pt(\hskip 1.49994pty\hskip 1.49994pt)$.    Hence  in  this case $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.49994pt)$  also  has  two  nearly  colorful  doors.    $\blacksquare$

8. Scarf’s  combinatorial  theorem.   *There  is  at  least  one  colorful  room.   Moreover ,   the number of  such  rooms  is  odd.   *

**An  informal  proof **.    The following  informal  explanation of  the proof  is  based  on  Scarf’s  outline  (see  [$S\mathrm{c}_{\hskip 0.70004pt3}$],   p.  48).   Actually ,   the  terms  rooms  and  doors  were suggested  by  this outline,   although  Scarf  used  them only  informally .   Let  us  think  about $T$ and  its  rooms and doors  as  if  it  is  a  building .   The above  lemmas  tell  us all  what  we need  to know about  this building .   We are interested only  in colorful  and  nearly  colorful  rooms and  in  nearly  colorful  doors.   One can  imagine  that  all  other doors and  rooms are permanently  locked.

Our  plan  is  to enter  this building  through an outside door and  then follow rooms and doors,   never  turning  back.   An outside door  leads  to a  particular  room,   which  is  either colorful  or  has only  one other  unlocked  door .   This unlocked door  leads us  to another  room,   which  is  also either colorful  or  has only  one other  unlocked  door .   We will  continue  to explore  the building  in  this way  until  we enter a  colorful  room.   Since we do not  turn  back,   we cannot  pass  through  the same door or enter  the same room  twice  (a room  has only  two unlocked doors).   Potentially ,   our journey  may  end at  another outside door .   But  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  implies  that  the  types of  doors and  rooms stay  the same during our  journey ,   and  by  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  there  is  only  one outside door of  any  given  type.   Since  we cannot  return  to  this door and  there are only  finitely  many  rooms,   we will  eventually  reach a colorful  room.

We found a particular colorful  room.   If  there are other ,   then  the same argument  shows  that  they  are pairwise related  by  routes consisting of  nearly  colorful  rooms and doors of  the  type $i$,   where $i$ is  the  type of  the outside door used  to enter  the building .   Therefore  the number of  other colorful  rooms  is  even and  the  total  number of  colorful  rooms  is  odd.

The graphs of  rooms and doors.   In order  to  turn  the above informal  proof  in a formal  one,   it  is  convenient  to represent  journeys  through  the building  by  paths in appropriate graphs.   Let  us  fix  some element  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  and  restrict  our  attention  by  the  colorful  rooms  and  by  the nearly  colorful  rooms and doors of  the  type $i$.   The relation

[TABLE]

between such  rooms and doors can  be encoded  in  terms of  a  graph  $G_{\hskip 0.70004pti}$,   having as  the vertices  the  colorful  rooms and  the nearly  colorful  rooms and doors of  the  type $i$.   Each  such  room  is  connected  by  an edge  to each of  its nearly  colorful  doors.   There are no other edges.   Clearly ,   Scarf’s  combinatorial  theorem  immediately  follows  from  the following  fact .

9. Theorem.   *The  graph  $G_{\hskip 0.70004pti}$  consists of  several  disjoint  paths and cycles.   With  one exception,   the endpoints  of  these paths  are  properly  colored  cells.   The only  exception  is  $(\hskip 1.00006pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$.   *

Proof**.**   By  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  there  is  only  one outside door  with  the  type $i$.   By  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  it  is  a door  of  exactly  one room,   which  is  nearly  colorful  and  has  the  type $i$  by  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.   Therefore  this outside door  is  an  endpoint  of  exactly  one edge.   Also  by  Lemma  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  every  colorful  room  has exactly  one nearly  colorful  door with  the  type $i$ and  hence  is  also an endpoint  of  exactly  one edge.   Lemmas  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  and  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  imply  that  every  internal  door  is  an  endpoint  of  two edges.   Finally ,   Lemmas  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  and  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  imply  that  every  nearly  colorful  room  is  an endpoint  of  two edges.   To sum  up,   every  vertex of  $G_{\hskip 0.70004pti}$  is  an endpoint  of  one or  two edges.   A vertex  is  an endpoint  of  only  one edge  if  and  only  if  it  is  either  a  colorful  room  or  the unique outside door  of  the  type $i$.   The  theorem  follows.    $\blacksquare$

Remarks.   The above proof  can  be easily  turned  into an algorithm for finding a  properly  colored  room.   In  this circle of  questions such  path-following  algorithms  originate  in  the work of  Scarf  [$S\mathrm{c}_{\hskip 0.70004pt2}$]  and are very  popular .   Such algorithms were actually  used,   already  by  Scarf ,   to compute approximations  to fixed  points of  self-maps of  a simplex .   See  [$S\mathrm{c}_{\hskip 0.70004pt2}$],   [$S\mathrm{c}_{\hskip 0.70004pt3}$].

2.   Brouwer’s  fixed  point  theorem

The standard simplex .   Let  us  turn  to  the  non-combinatorial  part  of  the proof  of  Brouwer’s  theorem.   Let  us  fix  a non-negative integer $n$,  and  let  $I\hskip 3.99994pt=\hskip 3.99994pt\{\hskip 1.99997pt0\hskip 0.50003pt,\hskip 3.00003pt1\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptn\hskip 1.99997pt\}$.   Let  us  number  the coordinates  in  $\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}$  by  elements of  $I$.   For  a  point  $x\hskip 1.99997pt\in\hskip 1.99997pt\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}$  let  $x_{\hskip 0.35002pti}$  be  the $i$-th coordinate of  $x$,   so  that   $x\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.99997ptx_{\hskip 1.04996pt0}\hskip 0.50003pt,\hskip 3.00003ptx_{\hskip 0.70004pt1}\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptx_{\hskip 0.70004ptn}\hskip 1.99997pt)$.   Let  $\Delta^{n}\hskip 3.99994pt\subset\hskip 3.99994pt\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}$  be  the standard $n$-simplex  defined  by  the equation  $x_{\hskip 1.04996pt0}\hskip 1.99997pt+\hskip 1.99997ptx_{\hskip 0.70004pt1}\hskip 1.99997pt+\hskip 1.99997pt\ldots\hskip 1.99997pt+\hskip 1.99997ptx_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pt1$  and  the inequalities  $x_{\hskip 0.70004pti}\hskip 1.99997pt\geqslant\hskip 1.99997pt0$  with  $i\hskip 1.99997pt\in\hskip 1.99997ptI$.   Let  $l\hskip 1.99997pt\geqslant\hskip 1.99997pt1$ be  another  integer  and  let  $T\hskip 3.99994pt=\hskip 3.99994ptT_{\hskip 0.35002ptl}$  be  the set  of  all  $x\hskip 1.99997pt\in\hskip 1.99997pt\Delta^{n}$  such that  every  $x_{\hskip 0.35002pti}$  is  an  integer  multiple  of  $1/\hskip 0.50003ptl$.   The set  $T$  will  serve as a discrete approximation  to  $\Delta^{n}$.

The  linear orders on  $T$.   For each  $i\hskip 1.99997pt\in\hskip 1.99997ptI$   let us choose a linear order  $<_{\hskip 0.70004pti}$  on  $T$  such  that

[TABLE]

for every  $x\hskip 0.50003pt,\hskip 3.00003pty\hskip 1.99997pt\in\hskip 3.00003ptT$  (obviously ,  such orders exist ).

10. Theorem.   Let  $\sigma\hskip 1.99997pt\subset\hskip 3.00003ptT$  and  $C\hskip 1.99997pt\subset\hskip 3.00003ptI$.   If   $\sigma$  is  dominant  with  respect  to  $C$,   then

[TABLE]

*for  every   $x\hskip 0.50003pt,\hskip 3.00003pty\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  and  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  and  $x_{\hskip 0.70004pti}\hskip 3.99994pt<\hskip 3.99994pt(\hskip 1.00006ptn\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.00006pt)\hskip 0.50003pt/\hskip 0.50003ptl$  for  every   $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  and  $i\hskip 1.99997pt\in\hskip 1.99997ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   *

Proof**.**   For each  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  let  $m\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\min\nolimits_{\hskip 0.70004pti}\hskip 1.99997pt\sigma$.   Let  $m_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptm\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)_{\hskip 0.70004pti}$  be  the $i$th  coordinate of  $m\hskip 1.00006pt(\hskip 1.00006pti\hskip 1.49994pt)$  for  $i\hskip 1.99997pt\in\hskip 3.00003ptC$  and  let  $m_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt0$  for  $i\hskip 1.99997pt\in\hskip 1.99997ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   By  using  the  triangle inequality  and  the fact  that  $|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 1.99997pt\leqslant\hskip 1.99997ptn\hskip 1.99997pt+\hskip 1.99997pt1$,   we see  that  it  is  sufficient  to prove  that

[TABLE]

for  every  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  and  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.   The  inequalities  $0\hskip 1.99997pt\leqslant\hskip 1.99997ptx_{\hskip 0.70004pti}\hskip 1.99997pt-\hskip 1.99997ptm_{\hskip 0.70004pti}$  hold  by  the definition of  $m_{\hskip 0.70004pti}$.   As  the first  step  toward  the inequalities  $x_{\hskip 0.70004pti}\hskip 1.99997pt-\hskip 1.99997ptm_{\hskip 0.70004pti}\hskip 3.99994pt<\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 0.50003pt\left/\hskip 1.00006ptl\right.$,    let  us  prove  that

[TABLE]

If  this  is  not  the case,   then

[TABLE]

Since every  $m_{\hskip 0.35002pti}$  is  a  multiple of  $1/\hskip 0.50003ptl$,   This  implies  that  there exists a  point  $M\hskip 1.99997pt\in\hskip 1.99997ptT$  such that  $M_{\hskip 0.70004ptk}\hskip 3.99994pt\hskip 3.99994pt\geqslant\hskip 3.99994pt\hskip 3.99994ptm_{\hskip 0.70004ptk}\hskip 1.99997pt+\hskip 1.99997pt(\hskip 1.00006pt1/\hskip 0.50003ptl\hskip 1.00006pt)$  and  hence  $\min\nolimits_{\hskip 0.70004ptk}\hskip 1.00006pt\sigma\hskip 3.99994pt<_{\hskip 0.70004ptk}\hskip 3.99994ptM$  for every  $k\hskip 1.99997pt\in\hskip 1.99997ptC$.   The contradiction  with $\sigma$ being  dominant  with  respect  to  $C$  proves  (6).

Let  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  and  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.    If  $i\hskip 1.99997pt\in\hskip 3.00003ptC$,    then

[TABLE]

If   $i\hskip 1.99997pt\not\in\hskip 3.00003ptC$,    then  $m_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt0$  and  hence

[TABLE]

Together  with  (6)  these inequalities  imply  that  $x_{\hskip 0.70004pti}\hskip 1.99997pt-\hskip 1.99997ptm_{\hskip 0.70004pti}\hskip 3.99994pt<\hskip 3.99994pt|\hskip 1.99997ptC\hskip 1.99997pt|\hskip 0.50003pt\left/\hskip 1.00006ptl\right.$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.    $\blacksquare$

Continuous self-maps of  $\Delta^{n}$.   Now  we  turn  to  the final  part  of  the proof  of  Brouwer’s  theorem.   Let  $f\hskip 1.00006pt\colon\hskip 1.00006pt\Delta^{n}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\Delta^{n}$  be a continuous map.   Recall  that  $T\hskip 3.99994pt=\hskip 3.99994ptT_{\hskip 0.35002ptl}$  depends on  $l$.    If

[TABLE]

then  $x_{\hskip 1.04996pt0}\hskip 1.99997pt+\hskip 1.99997ptx_{\hskip 0.70004pt1}\hskip 1.99997pt+\hskip 1.99997pt\ldots\hskip 1.99997pt+\hskip 1.99997ptx_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pty_{\hskip 1.04996pt0}\hskip 1.99997pt+\hskip 1.99997pty_{\hskip 0.70004pt1}\hskip 1.99997pt+\hskip 1.99997pt\ldots\hskip 1.99997pt+\hskip 1.99997pty_{\hskip 0.70004ptn}\hskip 3.99994pt=\hskip 3.99994pt1$  and  hence  $y_{\hskip 0.70004pti}\hskip 3.99994pt\geqslant\hskip 3.99994ptx_{\hskip 0.70004pti}$  for some  $i\hskip 1.99997pt\in\hskip 3.00003ptI$.   Let  $c\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)$  be equal  to any  such $i$.   This rule defines  a coloring of  $T_{\hskip 0.35002ptl}$,   i.e.  a map  $c\hskip 1.00006pt\colon\hskip 1.00006ptT_{\hskip 0.35002ptl}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptI$.

By  Theorem  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  for every  $l$  there  exists  a  properly  colored  cell,   i.e.  a  pair  $(\hskip 1.49994pt\sigma_{\hskip 0.35002ptl}\hskip 1.00006pt,\hskip 1.99997ptC_{\hskip 0.70004ptl}\hskip 1.49994pt)$  such  that  $C_{\hskip 0.70004ptl}$  is  non-empty ,  $\sigma_{\hskip 0.35002ptl}\hskip 3.99994pt\subset\hskip 3.99994ptT_{\hskip 0.35002ptl}$  is  dominant  with  respect  to  $C_{\hskip 0.70004ptl}$,   and  $C_{\hskip 0.70004ptl}\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma_{\hskip 0.35002ptl}\hskip 1.49994pt)$.   By  Theorem  2.   Brouwer’s  fixed  point  theorem  the diameter of  the sets  $\sigma_{\hskip 0.35002ptl}$  tends  to [math] when  $l\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\infty$.   Therefore,   after  passing to a subsequence,   still denoted  by  $\sigma_{\hskip 0.35002ptl}$,   we can assume  that  all  elements of  $\sigma_{\hskip 0.70004ptl}$ converge to the same point  $z\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.99997ptz_{\hskip 1.04996pt0}\hskip 0.50003pt,\hskip 3.00003ptz_{\hskip 0.70004pt1}\hskip 0.50003pt,\hskip 3.00003pt\ldots\hskip 0.50003pt,\hskip 3.00003ptz_{\hskip 0.70004ptn}\hskip 1.99997pt)\hskip 1.99997pt\in\hskip 3.00003pt\Delta^{n}$  when  $l\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\infty$.   Let

[TABLE]

Since there  are  only  finitely  many  subsets of  $I$,   after  passing  to a  further subsequence we can assume  that  $C_{\hskip 0.70004ptl}\hskip 3.99994pt=\hskip 3.99994ptC$  for some non-empty  subset  $C\hskip 3.99994pt\subset\hskip 3.99994ptI$  independent  of  $l$.   Then  $C\hskip 3.99994pt=\hskip 3.99994ptC_{\hskip 0.70004ptl}\hskip 3.99994pt=\hskip 3.99994ptc\hskip 1.00006pt(\hskip 1.00006pt\sigma_{\hskip 0.35002ptl}\hskip 1.49994pt)$  for every  $l$  and  hence for every  $l$  and every  $i\hskip 1.99997pt\in\hskip 3.00003ptC$  there  is  a  point  $z\hskip 1.00006pt(\hskip 1.00006pti\hskip 0.50003pt,\hskip 1.99997ptl\hskip 1.99997pt)\hskip 1.99997pt\in\hskip 3.00003pt\sigma_{\hskip 0.70004ptl}$  such  that  $c\hskip 1.00006pt(\hskip 1.49994ptz\hskip 1.00006pt(\hskip 1.00006pti\hskip 0.50003pt,\hskip 1.99997ptl\hskip 1.99997pt)\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pti$.   By  the choice of  the colorings $c$  passing  to  the limit  $l\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\infty$  shows  that  $w_{\hskip 0.70004pti}\hskip 3.99994pt\geqslant\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptC$.

At  the same  time  Theorem  5  implies  that  $x_{\hskip 0.70004pti}\hskip 1.99997pt<\hskip 1.99997pt(\hskip 1.00006ptn\hskip 1.99997pt+\hskip 1.99997pt1\hskip 1.00006pt)\hskip 0.50003pt/\hskip 0.50003ptl$   for  every  $x\hskip 1.99997pt\in\hskip 1.99997pt\sigma$  and  $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   By passing to the limit  $n\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\infty$  we conclude  that  $z_{\hskip 0.70004pti}\hskip 1.99997pt=\hskip 1.99997pt0$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$.   Therefore

[TABLE]

Since  $w_{\hskip 0.70004pti}\hskip 3.99994pt\geqslant\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 1.99997ptC$,   this equality  implies  that   $\displaystyle\sum\nolimits_{\hskip 2.10002pti\hskip 1.39998pt\in\hskip 1.39998ptC\vphantom{K^{K}}}\hskip 1.99997ptw_{\hskip 0.70004pti}\hskip 3.99994pt\hskip 3.99994pt\geqslant\hskip 3.99994pt\hskip 3.99994pt1$.   But

[TABLE]

It  follows  that  $w_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt0\hskip 3.99994pt=\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC$  and   $\displaystyle\sum\nolimits_{\hskip 2.10002pti\hskip 1.39998pt\in\hskip 1.39998ptC\vphantom{K^{K}}}\hskip 1.99997ptw_{\hskip 0.70004pti}\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt1\hskip 3.99994pt\hskip 3.99994pt=\hskip 3.99994pt\hskip 3.99994pt\sum\nolimits_{\hskip 2.10002pti\hskip 1.39998pt\in\hskip 1.39998ptC\vphantom{K^{K}}}\hskip 1.99997ptz_{\hskip 0.70004pti}$.

Since  $w_{\hskip 0.70004pti}\hskip 3.99994pt\geqslant\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 1.99997ptC$,   this equality  implies  that  $w_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptC$  also.   Therefore  $w_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994ptz_{\hskip 0.70004pti}$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  and  hence  $f\hskip 1.00006pt(\hskip 1.00006ptz\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptw\hskip 3.99994pt=\hskip 3.99994ptz$,   i.e.  $z$  is  a  fixed  point  of  the map  $f$.   This completes  the proof  of  Brouwer’s  fixed  point  theorem.    $\blacksquare$

3.   Scarf’s  proof  and  its  versions

Preferences and  utility  functions.   In  the context  of  the mathematical  economics and  the game  theory  the elements of  the set  $T$ can  be interpreted as goods,   and  the elements of  the set $I$ as  traders on a market .   From  this point  of  view,   the order  $<_{\hskip 0.70004pti}$  on  the set  of  goods reflects  the preferences of  the  trader  $i$.   It  is  quite natural  to  think  that  the preferences of  each  trader  $i$ arise from a  utility  function  $u_{\hskip 0.70004pti}\hskip 1.00006pt\colon\hskip 1.00006ptT\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbf{R}_{\hskip 1.39998pt+}\hskip 3.99994pt=\hskip 3.99994pt(\hskip 1.00006pt0\hskip 0.50003pt,\hskip 3.00003pt\infty\hskip 1.00006pt)$  such  that  the inequalities  $x\hskip 1.99997pt<_{\hskip 0.70004pti}\hskip 1.99997pty$  and  $u_{\hskip 0.70004pti}\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt<\hskip 1.99997ptu_{\hskip 0.70004pti}\hskip 1.00006pt(\hskip 1.00006pty\hskip 1.49994pt)$  are equivalent .   In any  case,   such  utility  functions always exist .   By  numbering  the  traders one can arrange utility  functions into a map  $u\hskip 1.00006pt\colon\hskip 1.00006ptT\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}_{\hskip 1.39998pt+}$  taking  $x\hskip 1.99997pt\in\hskip 3.00003ptT$  to  the point  $u\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)\hskip 1.99997pt\in\hskip 1.99997pt\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}_{\hskip 1.39998pt+}$  having  the utilities $u_{\hskip 0.70004pti}\hskip 1.00006pt(\hskip 1.00006ptx\hskip 1.49994pt)$ as  its coordinates.   By  perturbing  a  little  the utility  functions,   if  necessary ,   one can assume  that  $u$ is  injective,   and  then  identify  the set  $T$ with  its  image $u\hskip 1.00006pt(\hskip 1.49994ptT\hskip 1.49994pt)$.   Therefore,   without  any  loss of  generality  one can assume  that  $T$ is  a  finite subset  of  $\mathbf{R}^{\hskip 0.35002ptn\hskip 0.70004pt+\hskip 0.70004pt1}_{\hskip 1.39998pt+}$  and  the orders  $<_{\hskip 0.70004pti}$  are determined  by  the values of  coordinates.   For applications  to  Brouwer’s  fixed  point  theorem  the case of  $T\hskip 3.99994pt\subset\hskip 3.99994pt\Delta^{n}$  is  sufficient .   This  is  the framework of  Scarf’s  papers  [$S\mathrm{c}_{\hskip 0.70004pt1}$]  –  [$S\mathrm{c}_{\hskip 0.70004pt3}$].

Slack  vectors.   Scarf  enlarges  $T$  by  adding  to  it  vectors representing  the $(\hskip 0.50003ptn\hskip 1.00006pt-\hskip 1.00006pt1\hskip 0.50003pt)$-faces of  $\Delta^{n}$.   The face defined  by  the equation  $x_{\hskip 0.70004pti}\hskip 3.99994pt=\hskip 3.99994pt0$  is  represented  by

[TABLE]

the vector  with  the $i$th  coordinate equal  to [math] and other coordinates equal  to some real  number  $M_{\hskip 0.70004pti}$  bigger  than  the $i$th  coordinate of  every  vector  $x\hskip 1.99997pt\in\hskip 3.00003ptT$.   If  $T\hskip 3.99994pt\subset\hskip 3.99994pt\Delta^{n}$,   it  is  sufficient  to assume  that  $M_{\hskip 0.70004pti}\hskip 1.99997pt>\hskip 1.99997pt1$.   To stress an analogy  with  linear  programming ,   Scarf  calls  these vectors  slack  vectors.   Let  $\mathbb{I}$  be  the set  of  the slack  vectors.

It  is  convenient  to identify  the $i$th  trader  with  the $i$th  slack  vector  $s\hskip 1.49994pt(\hskip 1.00006pti\hskip 1.49994pt)$  for every  $i\hskip 1.99997pt\in\hskip 3.00003ptI$  and  therefore  identify  $I$  with  $\mathbb{I}$.   The enlarged  set  $T$  is  the union  $T\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$.   If  $M_{\hskip 0.70004pti}$  are pairwise different ,   then  the values of  coordinates define  linear  orders on  $T\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$.    A  subset  $X\hskip 3.99994pt\subset\hskip 3.99994ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  is  called  primitive   if  $|\hskip 1.99997ptX\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994ptn\hskip 1.99997pt+\hskip 1.99997pt1$  and  $X$  is  a  dominant  subset  of  $T\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  with  respect  to  the set  $I\hskip 3.99994pt=\hskip 3.99994pt\mathbb{I}$  of  all  orders.

By  the choice of  slack  vectors a  subset  $\sigma\hskip 3.99994pt\subset\hskip 3.99994ptT$  is  dominant  with  respect  to  $C\hskip 3.99994pt\subset\hskip 3.99994pt\mathbb{I}$  if  and  only  if  $\sigma\hskip 1.99997pt\cup\hskip 1.99997pt(\hskip 1.99997pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC\hskip 1.99997pt)$  is  a  primitive set .   Conversely ,  $X\hskip 3.99994pt\subset\hskip 3.99994ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  is  a  primitive subset  if  and  only  if  the intersection  $X\hskip 1.99997pt\cap\hskip 1.99997ptT$  is  dominant  with  respect  to  $\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptX$.   Therefore  the notions of  primitive and dominant  sets are equivalent  and  one can use either  of  them.

Scarf’s  main  lemma.   *Suppose  that  $X\hskip 3.99994pt\subset\hskip 3.99994ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  is  a  primitive set ,   and  $x\hskip 1.99997pt\in\hskip 3.00003ptX$.   Then either  $X\hskip 1.99997pt-\hskip 1.99997ptx\hskip 3.99994pt\subset\hskip 3.99994pt\mathbb{I}$,   or  there  is  a  unique  $y\hskip 1.99997pt\in\hskip 3.00003pt(\hskip 1.99997ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt)\hskip 3.99994pt\smallsetminus\hskip 3.99994ptX$  such  that  $X\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.99997pt+\hskip 1.99997pty$  is  a  primitive set .   For every  $i\hskip 1.99997pt\in\hskip 3.00003pt\mathbb{I}$  there  is  exactly  one  primitive set  containing  $\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$.   *

See  [$S\mathrm{c}_{\hskip 0.70004pt2}$],   Lemma  1.   The analogy  between  primitive sets and $n$-simplices of  a  triangulation of  $\Delta^{n}$  suggests  to state  this  lemma differently .   Let  us  call  a  subset  $Y\hskip 3.99994pt\subset\hskip 3.99994ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  almost  primitive   if  $|\hskip 1.99997ptY\hskip 1.99997pt|\hskip 3.99994pt=\hskip 3.99994ptn$  and  there  is  a  primitive set  containing  $Y$.   The set  $Y\hskip 3.99994pt=\hskip 3.99994ptX\hskip 1.99997pt-\hskip 1.99997ptx$  from  Scarf’s  main  lemma  is  almost  primitive,   and  hence we can  restate  this  lemma as follows.

Another  form  of  Scarf’s  main  lemma.   *Suppose  that  $Y\hskip 1.99997pt\subset\hskip 3.00003ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  is  an  almost  primitive set .   Then either   $Y\hskip 3.99994pt\subset\hskip 3.99994pt\mathbb{I}$,   or  $Y$  is  a subset  of  exactly  two primitive sets.   For every  $i\hskip 1.99997pt\in\hskip 3.00003pt\mathbb{I}$  the set  $\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  is  almost  primitive and  is  a subset  of  only  one primitive set .   *

By  the definitions,   if  $X\hskip 3.99994pt\subset\hskip 3.99994ptT\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}$  is  a  primitive set ,   then  $(\hskip 1.99997ptX\hskip 1.99997pt\cap\hskip 1.99997ptT\hskip 0.50003pt,\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptX\hskip 1.99997pt)$  is  a  room,   and  if  $X$  is  an  almost  primitive set ,   then  this  pair  is  a  door .   Moreover ,   if  $Y$  is  an almost  primitive set ,  $X$  is  a  primitive set ,   and  $Y\hskip 3.99994pt\subset\hskip 3.99994ptX$,   then  $(\hskip 1.99997ptY\hskip 1.99997pt\cap\hskip 1.99997ptT\hskip 0.50003pt,\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptY\hskip 1.99997pt)$  is  a  door of  the room  $(\hskip 1.99997ptX\hskip 1.99997pt\cap\hskip 1.99997ptT\hskip 0.50003pt,\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptX\hskip 1.99997pt)$.   Conversely ,   if  $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.00006pt)$ is  a  room,   then  $\sigma\hskip 1.99997pt\cup\hskip 1.99997pt(\hskip 1.99997pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptC\hskip 1.99997pt)$  is  a  primitive set .   The corresponding statement  for  the doors  is  only  a  little  less obvious.   By  Lemmas  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  and  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  every  door  is  a  door  of  a  room  (one can also see  this directly ).   It  follows  that  if  $(\hskip 1.00006pt\tau\hskip 0.50003pt,\hskip 1.99997ptD\hskip 1.00006pt)$ is  a  door ,   then  $\tau\hskip 1.99997pt\cup\hskip 1.99997pt(\hskip 1.99997pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptD\hskip 1.49994pt)$  is  contained  in  a  primitive set  and  hence  is  an  almost  primitive set .   So,   the notions of  primitive and  almost  primitive sets  are equivalent  to  the notions of  rooms and  doors,   and  Scarf’s  main  lemma  is  equivalent  to  Lemmas  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  and  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  together .

Primitive  sets and  rooms.   An obvious advantage of  working  with  slack  vectors  and  primitive sets  is  the fact  that  an almost  primitive set  is  always obtained  by  removing a point  from a primitive set .   In contrast ,   a door  can  be obtained  from a room $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.00006pt)$ either  by  removing a point  from $\sigma$ or  by  adding  a point  to $C$.   Still ,   traders and  goods are  inherently  different ,  and  it  is  only  natural  to keep  track of  this distinction,   as  it  is  done in  the notion of  a  room.

Scarf’s  proof  of  his combinatorial  theorem  uncovered a structure  hidden  in a  family  of  orders on a  given set,   namely ,   the collection of  primitive sets.   H.  Kuhn  [Ku]  interpreted  this structure as  the structure of  a  simplicial  complex .   The $n$-simplices of  this complex  are primitive sets and  Scarf’s  main  lemma  means  that  this complex  is  a  pseudo-manifold.   Sperner’s  lemma  and  its proofs naturally  extend  to  pseudomanifolds  and  Scarf’s  combinatorial  theorem  can  be proved  by  applying  such an extension  to  this pseudo-manifold.

By  keeping  the distinction  between  goods and  traders one can see an even  more rich combinatorial  structure  behind  the family  of  orders.   Namely ,   a  room $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.00006pt)$ can  be  thought  as  “belonging”  to  the face $C$ of  the abstract  simplex  having  $\mathbb{I}$  as  the set  of  vertices.   This  leads  to a  refined  version of  Kuhn’s  interpretation and  to a  proof  of  Scarf’s  combinatorial  theorem  using  Scarf’s  main  lemma as  the starting  point  and  then  proceeding  in  the same way as  in  combinatorial  or  homological  proofs  of  Sperner’s  lemma.   See  [$I_{\hskip 0.70004pt2}$],   Section  1.

Petri–Voorneveld  version.   In contrast  with  Scarf  [$S\mathrm{c}_{\hskip 0.70004pt2}$]  and  with  Section  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting,   H.  Petri  and  M.  Voorneveld  [PV]  start  with  fixing  a coloring  $c\hskip 1.00006pt\colon\hskip 1.00006ptT\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997ptI$.   In  the language of  the present  paper ,   they  work only  with  pairs $(\hskip 1.00006pt\sigma\hskip 0.50003pt,\hskip 1.99997ptC\hskip 1.00006pt)$ such  that  $|\hskip 1.99997ptC\hskip 1.99997pt\smallsetminus\hskip 1.99997ptc\hskip 1.00006pt(\hskip 1.49994pt\sigma\hskip 1.49994pt)\hskip 1.99997pt|\hskip 3.99994pt\leqslant\hskip 3.99994pt1$.   Petri  and  Voorneveld  call  them  candidates.   One can easily  check  that  every  such  pair  is  either a room or a door ,   but  other  rooms and  doors are missing .   Of  course,   such  pairs  are  sufficient  for  proving  Scarf  combinatorial  theorem  (called  in  [PV]  No-bullying  lemma  ),   but  the structure uncovered  by  Scarf  remains  hidden.   Perhaps,   this reflects  the intention  to eliminate geometry .

One has  to admit  that  the previous paragraph  is  not  quite faithful  to  [PV].   After  fixing  a coloring ,   Petri  and  Voorneveld  replace each color  (i.e.  each of  the orders)  by  as many  of  its  copies as  there are elements of  $T$  with  this color .   This  trick  allows  to  turn  the coloring $c$ into a  bijection.   Petri  and  Voorneveld  go one step  further and  use  this  bijective coloring  to identify  $T$  and  $I$.   This erases  the difference between  goods and  traders completely  (actually ,   Petri  and  Voorneveld  speak about  toys and  kids).   For  the present  author  this  identification  was  the main  stumbling  block  in  reading  [PV].

The  paths  to  primitive sets and colorful  rooms.   Let  $c\hskip 1.00006pt\colon\hskip 1.00006ptT\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbb{I}$  be  a coloring ,   and  let  us extend $c$  to a coloring  $T\hskip 1.99997pt\cup\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\hskip 1.99997pt\longrightarrow\hskip 1.99997pt\hskip 1.99997pt\mathbb{I}$  equal  to  the identity  on  $\mathbb{I}$.   There  is  no danger  in  denoting  this extension also by  $c$.   In  this  language  Scarf’s  combinatorial  theorem asserts  that  there  is  a  primitive set  $X$  such  that  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathbb{I}$.   In order  to prove  this,   Scarf’s  uses as  the main  tool  the operation of  replacing  $X$  by  $X\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.99997pt+\hskip 1.99997pty$  as  in  the main  lemma.   This  replaces  in  $X$  the element  $x$  by  a new element  $y$  and  is  called  a  replacement  step.   The proof  starts with  fixing a color  $i\hskip 1.99997pt\in\hskip 3.00003pt\mathbb{I}$  and considering  the set  $\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$.   Clearly ,   the set  $\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  is  almost  primitive and  by  the main  lemma  is  contained  in a unique primitive set  $X$.   If  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptI$,   then we are done.   Otherwise  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  and  one can apply  to  $X$  a sequence of  replacement  steps.   The elements  to replace are chosen  in such a way  that  after each  step either  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptI$  or  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$.   Scarf  proves  that  such sequence of  replacement  steps eventually  arrives  at  a  primitive set  $X$  such  that  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994ptI$,   when  it  stops.

One can split  each  replacement  step  into  two simpler steps.   Namely ,   instead  of  passing  from a  primitive set  $X$  to  $X\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.99997pt+\hskip 1.99997pty$  as in  the  main  lemma,   one can  first  pass from  $X$  to an almost  primitive set  $X\hskip 1.99997pt-\hskip 1.99997ptx$  and  then  pass from  $X\hskip 1.99997pt-\hskip 1.99997ptx$  to  $X\hskip 1.99997pt-\hskip 1.99997ptx\hskip 1.99997pt+\hskip 1.99997pty$.   Let  us  split  in  this way  every  replacement  step in  Scarf’s  sequence and add  $\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  at  the beginning .   The resulting  sequence alternates between almost  primitive and  primitive sets and  $c\hskip 1.49994pt(\hskip 1.49994ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.99997pt\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  for every  its  term  $X$  except  of  the  last  one.   The transformation  $Z\hskip 3.99994pt\longmapsto\hskip 3.99994pt(\hskip 1.99997ptZ\hskip 1.99997pt\cap\hskip 1.99997ptT\hskip 0.50003pt,\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptZ\hskip 1.99997pt)$  turns  this sequence  into a sequence rooms and  doors starting  with  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  and ending  with a colorful  room.   The condition  $c\hskip 1.49994pt(\hskip 1.99997ptX\hskip 1.49994pt)\hskip 3.99994pt=\hskip 3.99994pt\hskip 1.99997pt\mathbb{I}\hskip 1.99997pt-\hskip 1.99997pti$  implies  that  doors and  rooms  in  this sequence  are  nearly  colorful  of  type $i$,   except  the last  one.   Moreover ,   this sequence  is  a  path  in  the graph  $G_{\hskip 0.70004pti}$  connecting  $(\hskip 1.49994pt\varnothing\hskip 0.50003pt,\hskip 3.00003pt\{\hskip 1.49994pti\hskip 1.99997pt\}\hskip 1.49994pt)$  with a colorful  room.   But  Theorem  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  implies  that  there  is  only  one  such  path.   Hence  the  transformation  $Z\hskip 3.99994pt\longmapsto\hskip 3.99994pt(\hskip 1.99997ptZ\hskip 1.99997pt\cap\hskip 1.99997ptT\hskip 0.50003pt,\hskip 3.00003pt\mathbb{I}\hskip 1.99997pt\smallsetminus\hskip 1.99997ptZ\hskip 1.99997pt)$  turns  Scarf’s  sequence  into  the  path  to a colorful  room  from  Section  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting.   In other  words,   the arguments of  Section  1.   Scarf’s  combinatorial  theorem  in  an  abstract  setting  lead  to essentially  the same  the path  to a colorful  room  as  Scarf’s  ones.   Of  course,   the same  is  true with  respect  to  the arguments of  Petri  and  Voorneveld  [PV].