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

Let R(Ka,b, Kc,d) be the minimum number n so that any n-vertex simple undirected graph G contains a Ka,b or its complement G′ contains a Kc,d. We demonstrate constructions showing that R(K2,b, K2,d) > b + d + 1 for d ≥ b ≥ 2. We establish lower bounds for R(Ka,b, Ka,b) and R(Ka,b, Kc,d) using probabilistic methods. We define R′(a, b, c) to be the minimum number n such that any n-vertex 3-uniform hypergraph G(V, E), or its complement G′(V, E) contains a Ka,b,c. Here, Ka,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 Ka,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. We show that R′(1, 1, b) ≤ 2b + 1. We have also generated examples to show that R′(1, 1, 3) ≥ 6 and R′(1, 1, 4) ≥ 7.