A proof of a conjecture on Ramsey numbers $B(2,2,3)$

Abstract

The bipartite Ramsey number $B(n_1,n_2,\ldots,n_t)$ is the least positive integer $b$ such that, any coloring of the edges of $K_{b,b}$ with $t$ colors will result in a monochromatic copy of $K_{n_i,n_i}$ in the $i-$th color, for some $i$, $1\leq i\leq t$. In this paper we obtain the exact values of bipartite Ramsey numbers $B(2,2,3)$. In particular, we prove the conjecture of Radziszowski at al. aobut $B(2,2,3)$ which was introduced in 2015. In fact we prov that $B(2,2,3)=17$.