The s-bipartite Ramsey numbers involving $K_{2,3}$ and $K_{3,3}$

Abstract

A complete bipartite graph with the number of two partitions s and t is denoted by $Ks,t$. For a positive integer s and two bipartite graphs G and H, the s-bipartite Ramsey number $B{R}_s(G,H)$ of G and H is the smallest integer t such that every 2-coloring of the edges of $K_{s,t}$ contains the a copy of G with the first color or a copy of H with the second color. In this paper, by using an integer linear program and the solver Gurobi Optimizer 8.0, we determine all the exact values of $B{R}_s(K_{2,3},K_{3,3})$ for all possible $s$. More precisely, we show that $B{R}_s(K_{2,3},K_{3,3})=13$ for $s$ $\in$ {8,9}, $B{R}_s(K_{2,3},K_{3,3})=12$ for $s\in \{10,11\}$, $B{R}_s(K_{2,3},K_{3,3})=10$ for $s=12$, $B{R}_s(K_{2,3},K_{3,3})=8$ for $s\in \{13,14\}$, $B{R}_s(K_{2,3},K_{3,3})=6$ for $s\in \{15,16,\dots ,20\}$, and $B{R}_s(K_{2,3},K_{3,3})=4$ for s ≥ 21. This extends the results presented in [Zhenming Bi, Drake Olejniczak and Ping Zhang, “The s-Bipartite Ramsey Numbers of Graphs $K_{2,3}$ and $K_{3,3}$” , Journal of Combinatorial Mathematics and Combinatorial Computing 106, (2018) 257-272].