On Generalized Folkman Numbers

Abstract

For graphs $G$, $G_1$ and $G_2$, let $G \to (G_1,G_2)$ signify that any red/blue edge-coloring of $G$ contains a red $G_1$ or a blue $G_2$, and let $f(G_1,G_2)$ be the minimum $N$ such that there is a graph $G$ of order $N$ with $\omega(G) = \max \{\omega(G_1),\omega(G_2)\}$ and $G \to (G_1,G_2)$. It is shown that $c_1(n/\!\log n)^{(m+1)/2} \leq f(K_m,K_{n,n}) \leq c_2 n^{m-1}$, where $c_i = c_i(m) \gt 0$ are constants. In particular, $cn^2/\log n \leq f(K_3,K_{n,n}) \leq 2n^2+2n-1$. Moreover, $f(K_m,T_n) \leq m^2(n-1)$ for all $n \geq m \geq 2$, where $T_n$ is a tree on $n$ vertices.