Abstract
The bipartite Ramsey number B(n1,n2,…,nt) is the least positive integer b, such that any coloring of the edges of Kb,b with t colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1≤i≤t. The values B(2,5)=17, B(2,2,2,2)=19 and B(2,2,2)=11 have been computed in several previously published papers. In this paper, we obtain the exact values of the bipartite Ramsey number B(2,2,3). In particular, we prove the conjecture on B(2,2,3) which was proposed in 2015—in fact, we prove that B(2,2,3)=17.
Highlights
The bipartite Ramsey number B(n1, n2, . . . , nt ) is the least positive integer b, such that any coloring of the edges of Kb,b with t colors will result in a monochromatic copy of Kni,ni in the i-th color, for some i, 1 ≤ i ≤ t
In part (b) of Theorem 4, we showed that ≤ n = ∑ | NG g ( xi ) ∩ Y1 | ≤
We prove that in any 3-edge coloring of K17,17 (say ( Gr, G b, G g ), where K2,2 * Gr, K2,2 * G b ), if there exists a vertex of V (K ), such that | NG g ( x )| = 9 and
Summary
The bipartite Ramsey number B(n1 , n2 , . The Zarankiewicz number z(Km,n , t) is defined as the maximum number of edges in any subgraph G of the complete bipartite graph Km,n , such that G does not contain Kt,t as a subgraph. We intend to get the exact value of the multicolor bipartite Ramsey numbers B(2, 2, 3). Let G be a graph with vertex set. We use [ X, Y ] to denote the set of edges between the bipartition ( X, Y ) of G. Let G = ( X, Y ) be a bipartite graph and Z ⊆ X or Z ⊆ Y, the degree sequence of Z denoted by DG ( Z ) =
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