Ramsey Goodness and Beyond

In a seminal paper from 1983, Burr and Erdos started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde

Large joints in graphs

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the the r-partite Turan graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.Comment: 9 page

Ramsey goodness of paths

Given a pair of graphs G and H, the Ramsey number R(G, H) is the smallest N such that every red-blue coloring of the edges of the complete graph KN contains a red copy of G or a blue copy of H. If graph G is connected, it is well known and easy to show that R(G, H) ≥ (|G|−1)(χ(H)−1)+σ(H), where χ(H) is the chromatic number of H and σ the size of the smallest color class in a χ(H)- coloring of H. A graph G is called H-good if R(G, H) = (|G| − 1)(χ(H) − 1) + σ(H). The notion of Ramsey goodness was introduced by Burr and Erd˝os in 1983 and has been extensively studied since then. In this short note we prove that n-vertex path Pn is H-good for all n ≥ 4|H|. This proves in a strong form a conjecture of Allen, Brightwell, and Skokan

Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.Comment: 26 page

On the Ramsey number of the triangle and the cube

The Ramsey number r(K 3,Q n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K 3,Q n )=2 n+1−1 for every n∈ℕ, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K 3,Q n )⩽7000·2 n . Here we show that r(K 3,Q n )=(1+o(1))2 n+1 as n→∞

Disproof of a conjecture of Conlon, Fox and Wigderson

For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$ triangles all sharing a common edge. Recently, Conlon, Fox and Wigderson (2023) conjecture that for any $0<\alpha<1$, the random lower bound $r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+o(n)$ would not be tight. In other words, there exists some constant $\beta=\beta(\alpha)>0$ such that $r(B_{\lceil\alpha n\rceil},B_n)\ge (\sqrt{\alpha}+1)^2n+\beta n$ for all sufficiently large $n$. This conjecture clearly holds for every $\alpha< 1/6$ from an early result of Nikiforov and Rousseau (2005), i.e., for every $\alpha< 1/6$ and large $n$, $r(B_{\lceil\alpha n\rceil},B_n)=2n+3$. We disprove the conjecture of Conlon et al. (2023). Indeed, we show that the random lower bound is asymptotically tight for every $1/4\leq \alpha\leq 1$. Moreover, we show that for any $1/6\leq \alpha\le 1/4$ and large $n$, $r(B_{\lceil\alpha n\rceil}, B_n)\le\left(\frac 32+3\alpha\right) n+o(n)$, where the inequality is asymptotically tight when $\alpha=1/6$ or $1/4$. We also give a lower bound of $r(B_{\lceil\alpha n\rceil}, B_n)$ for $1/6\le\alpha< \frac{52-16\sqrt{3}}{121}\approx0.2007$, showing that the random lower bound is not tight, i.e., the conjecture of Conlon et al. (2023) holds in this interval.Comment: 14 page