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Turán theorems for unavoidable patterns

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  26 April 2021

ANTÓNIO GIRÃO  and
BHARGAV NARAYANAN
ANTÓNIO GIRÃO
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT. Department of Mathematics, Rutgers University, Piscataway, NJ08854, U.S.A. e-mails: tzgiraoa@gmail.com, narayanan@math.rutgers.edu
BHARGAV NARAYANAN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT. Department of Mathematics, Rutgers University, Piscataway, NJ08854, U.S.A. e-mails: tzgiraoa@gmail.com, narayanan@math.rutgers.edu
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Abstract

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We prove Turán-type theorems for two related Ramsey problems raised by Bollobás and by Fox and Sudakov. First, for t ≥ 3, we show that any two-colouring of the complete graph on n vertices that is δ-far from being monochromatic contains an unavoidable t-colouring when δ ≫ n−1/t, where an unavoidable t-colouring is any two-colouring of a clique of order 2t in which one colour forms either a clique of order t or two disjoint cliques of order t. Next, for t ≥ 3, we show that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament when δ ≫ n−1/[t/2], where an unavoidable t-tournament is the blow-up of a cyclic triangle obtained by replacing each vertex of the triangle by a transitive tournament of order t. Conditional on a well-known conjecture about bipartite Turán numbers, both our results are sharp up to implied constants and hence determine the order of magnitude of the corresponding off-diagonal Ramsey numbers.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05D10: Ramsey theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 423 - 442
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), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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