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

Abstract For bipartite graphs G 1 , G 2 , … , G k , the bipartite Ramsey number b ( G 1 , G 2 , … , 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 ≥ 3 , b ( C 2 t 1 , C 2 t 2 , C 2 t 3 , C 2 t 4 ) for 3 ≤ t i ≤ 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 ≤ t i ≤ 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.