Ramsey Numbers and the Size of Graphs

Abstract

For two graphs $H$ and $G$, the Ramsey number $r(H, G)$ is the smallest positive integer $n$ such that every red-blue edge coloring of the complete graph $K_n$ on $n$ vertices contains either a red copy of $H$ or a blue copy of $G$. Motivated by questions posed by Erdos and Harary, in this note we study how the Ramsey number $r(K_s, G)$ depends on the size of the graph $G$. For $s \geq 3$, we prove that for every $G$ with $m$ edges, $r(K_s,G) \geq c(m/\log m)^{(s+1)/(s+3)}$ for some positive constant $c$ depending only on $s$. This lower bound improves an earlier result of Erdos, Faudree, Rousseau, and Schelp, and it is tight up to a polylogarithmic factor when $s=3$. We also study the maximum value of $r(K_s,G)$ as a function of $m$.