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On the KŁR conjecture in random graphs

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Abstract

The KŁR conjecture of Kohayakawa, Łuczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.

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Authors and Affiliations

  1. Mathematical Institute, Oxford, OX1 3LB, UK

    D. Conlon

  2. Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, CB3 0WB, UK

    W. T. Gowers

  3. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel

    W. Samotij

  4. Trinity College, Cambridge, CB2 1TQ, UK

    W. Samotij

  5. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, D-20146, Hamburg, Germany

    M. Schacht

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  1. D. Conlon
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  2. W. T. Gowers
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  3. W. Samotij
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  4. M. Schacht
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Correspondence to D. Conlon.

Additional information

Research supported by a Royal Society University Research Fellowship.

Research supported by a Royal Society 2010 Anniversary Research Professorship.

Research supported in part by a Trinity College JRF.

Research supported by the Heisenberg programme of the DFG.

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Conlon, D., Gowers, W.T., Samotij, W. et al. On the KŁR conjecture in random graphs. Isr. J. Math. 203, 535–580 (2014). https://doi.org/10.1007/s11856-014-1120-1

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