Tripartite Ramsey number r t(K 2,3,K 2,3)

Abstract

A graph G is n - partite, n ≥ 1, if it is possible to partition the set of points V (G) into n subsets V 1, V 2, ..., V n (called partite sets) such that every element of the set of lines E(G) joins a point of Vi to a point of V j, i ≠ j. For n = 2, and n = 3 such graphs are called bipartite graph, and tripartite graph respectively. A complete n-partite graph G is an n-partite graph with the added property that if u ∈ V i and v ∈ V j, i ≠ j, then the line uv ∈ E(G). If |V i| = p i, then this graph is denoted by K p1,p2,... ,pn. for the complete tripartite graph K s,s,s with the number of points p = 3s, let each line of the graph has either red or blue color. The smallest number s such that K s,s,s always contains K m,n with all lines of K m,n have one color (red or blue) is called tripartite Ramsey number and denoted by r t(K m,n,K m,n). In this paper, we show that rt(K 2,3,K 2,3) = 5.