Scarf's theorems, simplices, and oriented matroids

TL;DR

This paper revisits Scarf's fixed point theorem and Sperner's lemma, extending them to colorings with colors from oriented matroids and providing new proofs using combinatorial topology.

Contribution

It generalizes Scarf's results to oriented matroids and offers classical topological proofs, broadening the applicability of these combinatorial theorems.

Findings

Generalization of Scarf's theorem to oriented matroid colorings

New proofs of classical theorems using combinatorial topology

Extension of Kannai's theorem

Abstract

In 1967 Herbert Scarf suggested a new proof of Brouwer fixed point theorem based on a surprising analogue of Sperner's lemma. This analogue was motivated by Scarf's work in game theory and mathematical economics. Moreover, Scarf proved a much general version of Sperner's lemma dealing with colorings by vectors. The present paper begins by revisiting Scarf's ideas from the point of view of the basic theory of simplicial cochains in the spirit of author's papers arXiv:1909.00940 and arXiv:2012.13104. After this we get to the main new results of the paper, namely, to a generalization of Scarf results to colorings with colors belonging to an oriented matroid. No knowledge of the theory of oriented matroids is assumed. In the last section we return to the original Scarf theorem and reprove it using even more classical methods of the combinatorial topology of Euclidean spaces. Also, we generalize a theorem of Kannai.