Ramsey goodness of books revisited

Abstract

The Ramsey number r(G, H) is the minimum N such that every graph on N vertices contains G as a subgraph or its complement contains H as a subgraph. For integers n ≥ k ≥ 1, the k-book B k,n is the graph on n vertices consisting of a copy of K k , called the spine, as well as n − k additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph H on n vertices is called p-good if r(K p , H) = (p − 1)(n − 1) + 1. Nikiforov and Rousseau proved that if n is sufficiently large in terms of p and k, then B k,n is p-good. Their proof uses Szemerédi's regularity lemma and gives a tower-type bound on n. We give a short new proof that avoids using the regularity method and shows that every B k,n with n ≥ 2 k 10p is p-good.Using Szemerédi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on r(G, H) for several families of sparse graphs G and H as long as |V (G)| < δ|V (H)| for a small constant δ > 0. Using our techniques, we prove a new result of this type, showing that r(G, H) = (p − 1)(n − 1) + 1 when H = B k,n and G is a complete p-partite graph whose first p − 1 parts have constant size and whose last part has size δn, for some small absolute constant δ > 0.

Ramsey numbers of cycles versus general graphs

“…Summing (13), we obtain r ′ f ≥ r total (1 − ((16 × 10 4 )q −1/2 log 2 k) − 4f ≥ 3r total /4. We repeatedly apply Lemma 5.3 with…”

“…This connection to Turán's theorem highlights how Ramsey goodness results can generalise other results in graph theory. See [13,19,22,24,25] and their references for more recent progress in the area of Ramsey goodness, as well as the survey of Conlon, Fox and Sudakov [9,Section 2.5].…”

Off-diagonal book Ramsey numbers

“…One of the central results in the field of Ramsey goodness is due to Nikiforov and Rousseau [19], who proved an extremely general theorem about when this lower bound is tight. As a very special case of their theorem, one has the following result; see also [15] for a new proof with better quantitative bounds.…”

“…Thus, if we pick j ∈ S randomly, then E[d B (W j , A i )] = p ± ε 2 . Therefore, if we first sample j ∈ S randomly and then pick a random blue K k inside W j , then Lemma 2.3 implies Specifically, for every δ > 0, there is some θ > 0, such that, in any (p, θ)-quasirandom coloring of E(K N ), On the one hand, we have that if δ ≥ 1/N, then Moreover, the results of [15] show that 1/c 0 is at most single-exponential in a power of k.…”

A note on the uniformity threshold for Berge hypergraphs

“…Nikiforov and Rousseau [15] showed that the generalized book B p,q (m) is q-good for m large enough. The threshold on m was improved in [6].…”