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Density theorems for bipartite graphs and related Ramsey-type results

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Abstract

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements.

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Authors and Affiliations

  1. Department of Mathematics Princeton, 68 Washington Road, Princeton, NJ, 08544, USA

    Jacob Fox

  2. Department of Mathematics, UCLA, Math Sciences 7911, Los Angeles, CA, 90095, USA

    Benny Sudakov

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  1. Jacob Fox
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  2. Benny Sudakov
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Correspondence to Benny Sudakov.

Additional information

Research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship.

Research supported in part by NSF CAREER award DMS-0812005 and by USA-Israeli BSF grant.

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Fox, J., Sudakov, B. Density theorems for bipartite graphs and related Ramsey-type results. Combinatorica 29, 153–196 (2009). https://doi.org/10.1007/s00493-009-2475-5

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