A note on Ramsey numbers for minors

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 $$R_h(k; \ell) = (2\beta+o(1)) \ell k \sqrt{\log_2 k}.$$