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Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

Authors:	Johannes H Hattingh Ernst J Joubert

Institution:	1. Department of Mathematics, East Carolina University, Greenville, NC 27858, USA;2. Department of Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa

Abstract:	For bipartite graphs $G_{1}, G_{2}, \dots, G_{k}$ , the bipartite Ramsey number $b (G_{1}, G_{2}, \dots, G_{k})$ is the least positive integer $b$ so that any coloring of the edges of $K_{b, b}$ with $k$ colors will result in a copy of $G_{i}$ in the $i$ th color for some $i$ . In this paper, our main focus will be to bound the following numbers: $b (C_{2 t_{1}}, C_{2 t_{2}})$ and $b (C_{2 t_{1}}, C_{2 t_{2}}, C_{2 t_{3}})$ for all $t_{i} \geq 3,$ $b (C_{2 t_{1}}, C_{2 t_{2}}, C_{2 t_{3}}, C_{2 t_{4}})$ for $3 \leq t_{i} \leq 9,$ and $b (C_{2 t_{1}}, C_{2 t_{2}}, C_{2 t_{3}}, C_{2 t_{4}}, C_{2 t_{5}})$ for $3 \leq t_{i} \leq 5 .$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.

Keywords:	Bipartite graph Ramsey Cycle
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