Geodesic
The $m$-bipartite Ramsey number $BR_m(H_1,H_2)$
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 893-911

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In a (G^1,G^2) coloring of a graph G, every edge of G is in G^1 or G^2. For two bipartite graphs H_1 and H_2, the bipartite Ramsey number BR(H_1, H_2) is the least integer b≥ 1, such that for every (G^1, G^2) coloring of the complete bipartite graph K_b,b, results in either H_1⊆ G^1 or H_2⊆ G^2. As another view, for bipartite graphs H_1 and H_2 and a positive integer m, the m-bipartite Ramsey number BR_m(H_1, H_2) of H_1 and H_2 is the least integer n (n≥ m) such that every subgraph G of K_m,n results in H_1⊆ G or H_2⊆G. The size of m-bipartite Ramsey number BR_m(K_2,2, K_2,2), the size of m-bipartite Ramsey number BR_m(K_2,2, K_3,3) and the size of m-bipartite Ramsey number BR_m(K_3,3, K_3,3) have been computed in several articles up to now. In this paper we determine the exact value of BR_m(K_2,2, K_4,4) for each m≥ 2.
Keywords: Ramsey numbers, bipartite Ramsey numbers, complete graphs, $m$-bipartite Ramsey number 
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     title = {The $m$-bipartite {Ramsey} number $BR_m(H_1,H_2)$},
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Rowshan, Yaser. The $m$-bipartite Ramsey number $BR_m(H_1,H_2)$. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 893-911. http://geodesic.mathdoc.fr/item/DMGT_2024_44_3_a4/