# 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.

## Key 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.