On a Ramsey-type problem of Erdős and Pach

Abstract

In this paper we show that there exists a constant C > 0 such that for any graph G on Ck ln k vertices either G or its complement G has an induced subgraph on k vertices with minimum degree at least 1 2 (k − 1). This affirmatively answers a question of Erdős and Pach from 1983.

Discrepancy and large dense monochromatic subsets

“…Erdős and Pach [9] found that the quasi-Ramsey numbers undergo a dramatic change in growth in k in a narrow window around c = 1/2: if c < 1/2 then they have linear growth, while if c > 1/2 they have singly exponential growth. They developed a fairly precise understanding of the transition at the point c = 1/2 -the present authors together with Pach [17] and with Long [15] have recently refined this.…”

“…Here we give a proof of Theorem 1 and discuss some consequences of it. Our proof of Theorem 1 is based on the proof of [15,Theorem 2], which in turn is inspired by a method of Erdős and Pach [9].…”

“…Indeed, from (14) we can directly argue that m is bounded by a constant depending on q times k − 1. This means we do not need (15), which requires that ρ i is not too small in terms of k or C(ν). So we have the following corollary, which in particular implies that for any graph G on n vertices, g(G) = Ω(n/ ln n), partly answering the question of Falgas-Ravry, Markström, and Verstraëte [13].…”

“…Remark. By adapting some results in [15], which are based on discrepancy results of Spencer [23] and Lovász, Spencer and Vesztergombi [18], one can deduce from Theorem 1 that there is a set S of size exactly k which has minimum degree at least ρ i (k − 1) plus a constant times (k − 1)/ ln k. We leave the details to the reader.…”

A Precise Threshold for Quasi-Ramsey Numbers

“…(k) = Ω k (k−1) (k) = O(k 2 ),but what is the correct behaviour of R * (k−1) (k)? Note added: subsequent to the present work, three of the authors have improved the upper bound to O(k ln 2 k)[11].…”