Exactly Colored Complete Subgraphs of Infinite Graphs
Žarko Ranđelović
TL;DR
The paper investigates Erickson's conjecture on the existence of infinite exactly-m-colored subgraphs in infinite complete graphs under exact c-edge-colorings. It proves that for all sufficiently large m, the statement P(c,m) fails for every c>m by constructing finite obstructions in two parameter regimes: c significantly smaller than m and c comparable to m, using prime-number-theoretic constructions and probabilistic coloring methods. Together with Stacey and Weidl’s results, this reduces Erickson’s conjecture to a finite number of remaining cases. The work advances the understanding of Ramsey-type colorings and provides methods to rule out exact-color subgraphs, moving toward a complete resolution via finite verification.
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.