Related papers: Ramsey goodness of trees in random graphs

For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for…

R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every…

For graph $G$, a connected graph $H$ of order $n$ is said to be $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let…

Let $mH$ be the graph formed by $m$ vertex-disjoint copies of a graph $H$. Let $G \to (H)_r$ denote that, in any $r$-colouring of the edges of $G$, there exists a monochromatic copy of $H$. In 1975, Burr, Erd\H{o}s, and Spencer showed that…

Let $R(C_n)$ be the Ramsey number of the cycle on $n$ vertices. We prove that, for some $C > 0$, with high probability every $2$-colouring of the edges of $G(N,p)$ has a monochromatic copy of $C_n$, as long as $N\geq R(C_n) + C/p$ and $p…

In 1983, Burr and Erd\H{o}s initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erd\H{o}s, in which the bounds on the parameters are of tower type since…

Let $C_{n}$ be a cycle of length $n$. As an application of Szemer\'{e}di's regularity lemma, {\L}uczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that…

A well-known result of R\"odl and Ruci\'nski states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges…

More than thirty years ago, Erd\H{o}s, Faudree, Rousseau, and Schelp posed a fundamental question in extremal graph theory: What is the optimal constant $c_k$ such that $r(C_{2k+1}, G) \le c_k m$ for any graph $G$ with $m$ edges and no…

A sequence of nonnegative integers $\pi =(d_1,d_2,...,d_n)$ is graphic if there is a (simple) graph $G$ of order $n$ having degree sequence $\pi$. In this case, $G$ is said to realize or be a realization of $\pi$. Given a graph $H$, a…

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show…

A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ denotes the size of the smallest color class in a $\chi(H)$-coloring of $H$. In Ramsey theory, it is an interesting problem to study whether a graph…

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2…

We prove that for all epsilon>0 there are c>0 and n_0 such that for all n>n_0 the following holds. For any two-colouring of the edges of $K_{n,n,n}$ one colour contains copies of all trees T of order t<(3-epsilon)n/2 and with maximum degree…

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 $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it…

Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number…

For any countably infinite graph $G$, Ramsey's theorem guarantees an infinite monochromatic copy of $G$ in any $r$-coloring of the edges of the countably infinite complete graph $K_\mathbb{N}$. Taking this a step further, it is natural to…

A celebrated result of R\"odl and Ruci\'nski states that for every graph $F$, which is not a forest of stars and paths of length $3$, and fixed number of colours $r\ge 2$ there exist positive constants $c, C$ such that for $p \leq…

Our main result is a robust generalisation of the Cockayne-Lorimer theorem on the multicolour Ramsey number of matchings. It is moreover a generalisation of the transference generalisation of Cockayne-Lorimer, which (informally) says that…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…