Turán theorems for unavoidable patterns

António Girão,Bhargav Narayanan

doi:10.1017/s030500412100027x

Abstract

AbstractWe 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.

Highlights

The starting point of Ramsey theory, namely Ramsey’s theorem [15], is the assertion that given any natural number t ∈ N, every two-colouring of the complete graph Kn on n vertices contains a monochromatic copy of Kt for all large enough n ∈ N; the asymptotic behaviour of the smallest such integer, namely the Ramsey number R(t), has been the subject of intense scrutiny through the past seventy or so years
A priori, one cannot expect to find any non-monochromatic patterns in a given twocolouring of a complete graph, since the colouring in question might itself be monochromatic
Our main contribution in this paper was to pin down the order of magnitude of the Ramsey numbers C(t, δ) and D(t, δ) for fixed t ∈ N as δ → 0

Summary

Introduction

The starting point of Ramsey theory, namely Ramsey’s theorem [15], is the assertion that given any natural number t ∈ N, every two-colouring (of the edges, here and elsewhere) of the complete graph Kn on n vertices contains a monochromatic copy of Kt for all large enough n ∈ N; the asymptotic behaviour of the smallest such integer, namely the Ramsey number R(t), has been the subject of intense scrutiny (see [3, 5, 7, 17], for example) through the past seventy or so years. For each integer t ≥ 3, there exists a C = C(t) > 0 such that any two-colouring of the complete graph on n ≥ C vertices that is Cn−1/t -far from being monochromatic contains an unavoidable t-colouring. Fox and Sudakov [8] showed that for any t ∈ N. and δ > 0, there exists a least integer D(t, δ) ∈ N such that any tournament on n vertices that is δ-far from being transitive contains an unavoidable t-tournament for all n ≥ D(t, δ).

Preliminaries

Colourings

Exceptional patterns

Conclusion