Size multipartite Ramsey numbers for small paths versus books

Abstract

Given $j \ge 2$,  for  graphs $G$ and $H$, the size Ramsey multipartite number $m_j(G, H)$ is defined as the smallest natural number $t$ such that any blue red coloring of the edges of the  graph $K_{j \times t}$, necessarily containes a red $G$ or a blue $H$ as subgraphs. Let the book with $n$ pages is defined as the graph $K_1 + K_{1,n}$ and denoted by $B_n$. In this paper, we  obtain the exact values of the size Ramsey numbers $m_j(P_3, H)$ for $j \ge 3$ where  $H$ is a book $B_n$. We also derive some upper and lower bounds for the size Ramsey numbers $m_j(P_4, H)$ where  $H$ is a book $B_n$.