Bounds for Bipartite Rainbow Ramsey Numbers

Abstract

Given bipartite graphs G and H, the bipartite rainbow Ramsey number BRR(G; H) is the minimum integer N such that any edge-coloring of $K_{N,N}$ with any number of colors contains either a monochromatic copy of G or a rainbow copy of H. It is known that BRR(G; H) exists if and only if G is a star or H is a forest consisting of stars. For fixed $t\ge 3$, $s\ge (t-1)!+1$ and large n, we shall show that $BRR(K_{t,s};K_{1,n})=\varTheta (n^t)$ and $BRR(K_{1,n};K_{t,t})=\varTheta (n)$. We also improve the known bounds for $BRR(C_{2m};K_{1,n})$, $BRR(K_{1,n};C_{2m})$, $BRR(B_{s,t};K_{1,n})$ and $BRR(K_{1,n};B_{s,t})$, where $B_{s,t}$ is a broom consisting of $s+t$ edges obtained by identifying the center of star $K_{1,s}$ with an end-vertex of a path $P_{1+t}$. Particularly, we have $BRR(C_{2m};K_{1,n})\ge (1-o(1))n^{m/(m-1)}$ for $m=2,3,5$ and large n.