A Proof of Atanassov's Conjecture and Other Generalizations of Sperner's Lemma

TL;DR

This paper provides an elementary proof of Atanassov's Conjecture, a generalization of Sperner's Lemma, and introduces further generalizations and a new related theorem, simplifying understanding of these combinatorial topological results.

Contribution

The paper offers a simpler proof of Atanassov's Conjecture, extends it to new generalizations, and proves a novel theorem akin to Sperner's Lemma.

Findings

Elementary proof of Atanassov's Conjecture

Generalizations of the conjecture presented

A new Sperner-like theorem proved

Abstract

A simple proof of Atanassov's Conjecture is presented. Atanassov's Conjecture is a generalization of Sperner's Lemma, a lemma which has been used to prove Brouwer's Fixed Point Theorem, among other fixed point theorems. The proof of Atanassov's Conjecture is based on the Brouwer Degree of maps and is extremely elementary. It is much simpler than the original proofs given for the conjecture and provides some insight into the nature of the conjecture. Furthermore, a generalization of the conjecture is presented and finally a new theorem, similar to the original Sperner Lemma, is proved.