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On the Ramsey-Turán numbers of graphs and hypergraphs

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Abstract

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT t (n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α t (G) ≤ f(n), where α t (G) is the maximum number of vertices in a K t -free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT t (n,K s , o(n)), among them finding the minimum ℓ such that RT t (n,K t+ℓ , o(n)) = Ω(n 2), where it is easy to see that RT t (n,K t+1, o(n)) = o(n 2). In this paper, we answer this question by proving that RT t (n,K t+2, o(n)) = Ω(n 2); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.

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Authors and Affiliations

  1. Mathematics Department, University of Illinois, Urbana-Champaign, Urbana, IL, 61821, USA

    József Balogh & John Lenz

  2. Department of Mathematics, University of California, San Diego, San Diego, CA, 92093, USA

    József Balogh

Authors
  1. József Balogh
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  2. John Lenz
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Corresponding author

Correspondence to József Balogh.

Additional information

This material is based upon work supported by NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grants 11067, 09072 and 08086, and OTKA Grant K76099.

Work supported by 2010 REGS Program of the University of Illinois and the National Science Foundation through a fellowship funded by the grant DMS 0838434 “EMSW21MCTP: Research Experience for Graduate Students”.

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Balogh, J., Lenz, J. On the Ramsey-Turán numbers of graphs and hypergraphs. Isr. J. Math. 194, 45–68 (2013). https://doi.org/10.1007/s11856-012-0076-2

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  • DOI: https://doi.org/10.1007/s11856-012-0076-2

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