Ramsey-type Theorems with Forbidden Subgraphs

Dedicated to the memory of Paul Erdős

A graph is called H-free if it contains no induced copy of H. We discuss the following question raised by Erdős and Hajnal. Is it true that for every graph H, there exists an such that any H-free graph with n vertices contains either a complete or an empty subgraph of size at least ? We answer this question in the affirmative for a special class of graphs, and give an equivalent reformulation for tournaments. In order to prove the equivalence, we establish several Ramsey type results for tournaments.

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Received August 22, 1999

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ID="*" Supported by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.

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ID="†" Supported by NSF grant CR-9732101, PSC-CUNY Research Award 663472, and OTKA-T-020914.

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ID="‡" Supported by TKI grant Stochastics@TUB, and OTKA-T-026203.

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Alon, N., Pach, J. & Solymosi, J. Ramsey-type Theorems with Forbidden Subgraphs. Combinatorica 21, 155–170 (2001). https://doi.org/10.1007/s004930100016

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  • AMS Subject Classification (2000) Classes:  05D10, 05D40, 05C20