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Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

Part of: Graph theory

Published online by Cambridge University Press:  19 November 2020

W. T. Gowers  and
Barnabás Janzer
W. T. Gowers
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK
Barnabás Janzer*
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WB, UK
*
*Corresponding author. Email: bkj21@cam.ac.uk
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Abstract

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Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of Kr such that no Kr is rainbow.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 4 , July 2021 , pp. 591 - 608
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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