Size multipartite Ramsey numbers for small paths versus books

Chula J. Jayawardene (University of Colombo Colombo Sri Lanka)
Jayampathy Ratnayake (Faculty of Engineering Sri Lanka Institute of Information Technology Malambe Sri Lanka.)

Indonesian Journal of Combinatorics
Vol 1, No 1 (2016)

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 $B_n$.