Lower bounds for Ramsey numbers for complete bipartite and 3-uniform tripartite subgraphs
Vol. 17, no. 6, pp. 671-688, 2013. Regular paper.
Abstract 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 db ≥ 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,Ec) contains a Ka,b,c. Here, Ka,b,c is defined as the complete tripartite 3-uniform hypergraph with vertex set ABC, 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}, uA, vB and wC. 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.
Keywords: Ramsey number, bipartite graph, local lemma, probabilistic method, r-uniform hypergraph.
Submitted: April 2013.
Reviewed: September 2013.
Revised: October 2013.
Accepted: October 2013.
Final: October 2013.
Published: November 2013.
Communicated by Subir Kumar Ghosh
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