Exactly Colored Complete Subgraphs of Infinite Graphs

TL;DR

This paper investigates the existence of exactly colored infinite subgraphs within infinite graphs, proving that for large enough subgraph sizes, the conjecture holds for all larger color counts, reducing the problem to finitely many cases.

Contribution

It demonstrates that for sufficiently large subgraph sizes, the conjecture is true for all larger numbers of colors, simplifying the verification process.

Findings

The conjecture holds for all large enough m and all c > m.

Reduces the problem to finitely many cases for verification.

Extends previous results by Stacey and Weidl.

Abstract

Given integers $m\le c$ and an exact $c$-coloring of the edges of a complete countably infinite graph (i.e. a coloring that uses exactly $c$ colors), must there be an infinite subgraph that is exactly $m$-colored? Using the Infinite Ramsey Theorem It is easy to show that the statement is true if $m=1,2$ or $c$. Erickson conjectured that it is false in all other cases. Stacey and Weidl proved that for each $m\ge 3$ there is some large enough $C(m)$ such that the conjecture is true for all pairs $(c,m)$ with $c>C(m)$. The main aim of this paper is to show that for all large enough $m$ the conjecture holds for all $c>m$. This reduces the number of cases needed to fully verify the conjecture to a finite number.