Tripartite Ramsey number r t(K 2,4, K 2,4)

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 V i 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 colour. The smallest number s such that K s,s,s always contains K m,n with all lines of K m,n have one colour (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 r t(K 2,4, K 2,4) = 7. © 2012 by the Mathematical Association of Thailand. All rights reserved.