Lower Bounds for Ramsey Numbers for Complete Bipartite and 3-Uniform Tripartite Subgraphs

Abstract

In this paper we establish lower bounds of the number R′(a,b) so that any bicoloring of the edges of the complete undirected graph K _n with n ≥ R′(a,b) vertices, always admits a monochromatic complete bipartite subgraph K _a,b , where a and b are natural numbers. We show that R′(2,b) > 2b + 1 for b ≥ 4. We establish a lower bound for R′(a,b) using the probabilistic method that improves the lower bound given by Chung and Graham [4]. Further, we also use Lovász’ local lemma to derive a better lower bound for R′(a,b). We define R′(a,b,c) be the minimum number n such that any n-vertex 3-uniform hypergraph G(V,E), or its complement G′(V,E ^c) contains a K _a,b,c . Here, K _a,b,c is defined as the complete tripartite 3-uniform hypergraph with vertex set A ∪ B ∪ C, where the A, B and C have a, b and c vertices respectively, and K _a,b,c has abc 3-uniform hyperedges {u,v,w}, u ∈ A, v ∈ B and w ∈ C. We derive lower bounds for R′(a,b,c) using probabilistic methods.