A note on Ramsey numbers for minors

TL;DR

This paper determines the minimal size of complete graphs needed to guarantee monochromatic minors of a given graph or clique under two-color and multi-color edge colorings, extending previous asymptotic results.

Contribution

It provides exact asymptotic formulas for Ramsey numbers for minors with two colors and extends to multiple colors, generalizing prior asymptotic bounds.

Findings

Exact asymptotic formula for R_h(F;2) for large graphs.

Asymptotic behavior of R_h(k; \, ext{colors}) for large k and \, ext{colors}.

Extension of previous bounds to more colors and general graphs.

Abstract

Let $R_h(k; \ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $K_k$-minor, in other words, a graph with Hadwiger number $k$, i.e., a graph that could be transformed into a clique $K_k$ on $k$ vertices via a sequence of edge contractions and vertex deletions. More generally, for a graph $F$ and integer $\ell$ let $R_h(F;\ell)$ be the smallest integer $n$ such that any edge coloring of a complete graph on $n$ vertices in $\ell$ colors results in a monochromatic $F$-minor. In 2001 Thomason and in 2005 Myers and Thomason asymptotically determined the extremal numbers for clique minors and $F$-minors, respectively. They found the respective explicitly computable leading constants $\beta=0.265656...$ and $\gamma(F)\cdot \beta$ for these extremal numbers. We determine $R_h(F;2)$ for every graph $F$ as $$R_h(F;2)=(\gamma(F)+o(1))|V(F)|\sqrt{\log_2(|V(F)|)},$$ where the $o(1)$-term tends to zero as $|V(F)|\rightarrow \infty$. In particular, $$R_h(k;2)=(1+o(1))k\sqrt{\log_2 k}.$$ When $\ell\gg k \gg 1$, we show that $$ R_h(k; \ell) = (2\beta+o(1)) \ell k \sqrt{\log_2 k}.$$