On the multicolor Ramsey number of a graph with m edges

Kathleen Johst; Yury Person

paper

On the multicolor Ramsey number of a graph with m edges

arXiv:1311.5471

Abstract

The multicolor Ramsey number $r_{k} (F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_{n}$ by $k$ colors there is a monochromatic copy of $F$ . In this short note we prove an upper bound on $r_{k} (F)$ for a graph $F$ with $m$ edges and no isolated vertices of the form $k^{6 k m^{2/3}}$ addressing a question of Sudakov [ Adv. Math. 227 (2011), no. 1, 601--609]. Furthermore, the constant in the exponent in the case of bipartite $F$ and two colors is lowered so that $r_{2} (F) \leq 2^{(1 + o (1)) 2 2 m}$ improving the result of Alon, Krivelevich and Sudakov [Combin. Probab. Comput. 12 (2003), no. 5--6, 477--494].

concluding remarks expanded