For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$. In 1977, Chvátal proved that for any integers $r,n,D \ge 2$, $K_N \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if and only if $N \ge rn+1$. We prove a random analogue of Chvátal's theorem for bounded degree trees, that is, we show that for each $r,D\ge 2$ there exist constants $C,C'>0$ such that if $p \ge C{n}^{-2/(r+2)}$ and $N \geq rn + C'/p$, then \[G(N,p) \rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)\] with high probability as $n\to \infty$. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erdős--Sós conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.
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.
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 graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, 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 Erdős in 1983 and has been extensively studied si…
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 is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is 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 Erdős in 1983 and has been extensively st…
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 is well known and easy to show that $R(G,H) \geq (|G|-1)(χ(H)-1)+σ(H)$, where $χ(H)$ is the chromatic number of $H$ and $σ(H)$ is 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 Erdős in 1983 and has been extensively st…
We say that a graph $G$ is Ramsey for $H_1$ versus $H_2$, and write $G \to (H_1,H_2)$, if every red-blue colouring of the edges of $G$ contains either a red copy of $H_1$ or a blue copy of $H_2$. In this paper we study the threshold for the event that the Erdős--Rényi random graph $G(N,p)$ is Ramsey for a clique versus a path. We show that $$G\big( (1 + \varepsilon) rn,p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 1)}$, and $$G\big( rn + t, p \big) \to (K_{r+1},P_n)$$ with high probability if $p \gg n^{-2 / (r + 2)}$ and $t \gg 1/p$. Both of these results are sharp (…
In 1983, Burr and Erdős initiated the study of Ramsey goodness problems.Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let $B_{k,n}$ be the book graph on $n$ vertices which consists of $n-k$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $H=K_p(a_1,\dots,a_{p})$ be the complete $p$-partite graph with parts of sizes $a_1,\dots,a_{p}$. Recently, avoiding use of the regularity lemma, Fox, He and Wigderson (2021) revisit several Ramsey go…
A celebrated result of Chvátal, Rödl, Szemerédi and Trotter states (in slightly weakened form) that, for every natural number $Δ$, there is a constant $r_Δ$ such that, for any connected $n$-vertex graph $G$ with maximum degree $Δ$, the Ramsey number $R(G,G)$ is at most $r_Δn$, provided $n$ is sufficiently large. In 1987, Burr made a strong conjecture implying that one may take $r_Δ= Δ$. However, Graham, Rödl and Ruciński showed, by taking $G$ to be a suitable expander graph, that necessarily $r_Δ> 2^{cΔ}$ for some constant $c>0$. We show that the use of expanders is essential: if we impose the…
The classical Ramsey theorem was generalized in two major ways: to the dual Ramsey theorem, by Graham and Rothschild, and to Ramsey theorems for trees, initially by Deuber and Leeb. Bringing these two lines of thought together, we prove the dual Ramsey theorem for trees. Galois connections between partial orders are used in formulating this theorem, while the abstract approach to Ramsey theory, we developed earlier, is used in its proof.
We estimate the Ramsey number r(T) = r(T,T) for various trees T, obtaining a precise value for r(T) for a large number of trees of diameter 3. Furthermore we prove that all trees of diameter 3 are Ramsey unsaturated as defined by Balister, Lehel, and Schelp in their article "Ramsey unsaturated and saturated graphs."
In this paper we study Ramsey numbers for trees of diameter 3 (bistars) vs., respectively, trees of diameter 2 (stars), complete graphs, and many complete graphs. In the case of bistars vs. many complete graphs, we determine this number exactly as a function of the Ramsey number for the complete graphs. We also determine the order of growth of the bipartite $k$-color Ramsey number for a bistar.
We show that the class of finite rooted binary plane trees is a Ramsey class (with respect to topological embeddings that map leaves to leaves). That is, for all such trees P,H and every natural number k there exists a tree T such that for every k-coloring of the (topological) copies of P in T there exists a (topological) copy H' of H in T such that all copies of P in H' have the same color. When the trees are represented by the so-called rooted triple relation, the result gives rise to a Ramsey class of relational structures with respect to induced substructures.
Given a positive integer $s$, the $s$-colour size-Ramsey number of a graph $H$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that, in any colouring of $E(G)$ with $s$ colours, there is a monochromatic copy of $H$. We prove that, for any positive integers $k$ and $s$, the $s$-colour size-Ramsey number of the $k$th power of any $n$-vertex bounded degree tree is linear in $n$. As a corollary we obtain that the $s$-colour size-Ramsey number of $n$-vertex graphs with bounded treewidth and bounded degree is linear in $n$, which answers a question rai…
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…
It is known that the connected graphs that are $p$-good for all $p$ are the trees. In this paper, we ask which ordered trees are order-$p$-good for all $p$.
An edge-colored graph $G$ is called rainbow if every edge of $G$ receives a different color. The anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. In this paper, we prove this conjecture. We also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of …
In this paper, we obtain upper bounds for the geometric Ramsey numbers of trees. We prove that $R_c(T_n,H_m)=(n-1)(m-1)+1$ if $T_n$ is a caterpillar and $H_m$ is a Hamiltonian outerplanar graph on $m$ vertices. Moreover, if $T_n$ has at most two non-leaf vertices, then $R_g(T_n,H_m)=(n-1)(m-1)+1$. We also prove that $R_c(T_n,H_m)=O(n^2m)$ and $R_g(T_n,H_m)=O(n^3m^2)$ if $T_n$ is an arbitrary tree on $n$ vertices and $H_m$ is an outerplanar triangulation with pathwidth 2. %Further, we prove a uniform polynomial upper bound for the geometric Ramsey numbers of caterpillars and we also give an upp…
We say that a graph $F$ can be embedded into a graph $G$ if $G$ contains an isomorphic copy of $F$ as a subgraph. Guo and Volkmann \cite{GV} conjectured that if $G$ is a connected graph with at least $n$ vertices and minimum degree at least $n-3$, then any tree with $n$ vertices and maximum degree at most $n-4$ can be embedded into $G$. In this paper, we give a result slightly stronger than this conjecture and obtain a sufficient and necessary condition that a tree with $n$ vertices and maximum degree at most $n-3$ can be embedded into a connected graph G with at least $n$ vertices and minimum…
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 at most n^c. This confirms a conjecture of Schelp.
We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-unif…
Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a function of both m and n. In the end, we refine the conjecture of Chen et al. by giving the condition of small maximum degree. For a tree Tn wit…
In this paper, we study Ramsey-type problems for directed graphs. We first consider the $k$-colour oriented Ramsey number of $H$, denoted by $\overrightarrow{R}(H,k)$, which is the least $n$ for which every $k$-edge-coloured tournament on $n$ vertices contains a monochromatic copy of $H$. We prove that $ \overrightarrow{R}(T,k) \le c_k|T|^k$ for any oriented tree $T$. This is a generalisation of a similar result for directed paths by Chvátal and by Gyárfás and Lehel, and answers a question of Yuster. In general, it is tight up to a constant factor. We also consider the $k$-colour directed Rams…
I will give a presentation of an abstract approach to finite Ramsey theory found in an earlier paper of mine. I will prove from it a common generalization of Deuber's Ramsey theorem for regular trees and a recent Ramsey theorem of Jasinski for boron tree structures. This generalization appears to be new. I will also show, in exercises, how to deduce from it the Milliken Ramsey theorem for strong subtrees.
Burr, Erdős, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 50m$. Combining this with a result of Faudree, Lawrenc…
The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for $k$-uniform hypergraphs. Analogous to the graph case, we consider the size-Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size-Ramsey numbers for hypergraphs are extremely difficult to determine, and many…
An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. Given any host graph $G$, the \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees in $G$, denoted by $r(G,t)$, is defined as the maximum number of colors in an edge-coloring of $G$ containing no $t$ edge-disjoint rainbow spanning trees. For any vertex partition $P$, let $E(P,G)$ be the set of non-crossing edges in $G$ with respect to $P$. In this paper, we determine $r(G,t)$ for all host graphs $G$: $r(G,t)=|E(G)|$ if there exists a partition $P_0$ with $|E(G)|-|E(P_0,G)|<t…
Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or …
Let $G$ be a complete multi-partite graph of order $n$. In this paper, we consider the anti-Ramsey number $ar(G,\mathcal{T}_{q})$ with respect to $G$ and the set $\mathcal{T}_{q}$ of trees with $q$ edges, where $2\le q\le n-1$. For the case $q=n-1$, the result has been obtained by Lu, Meier and Wang. We will extend it to $q<n-1$. We first show that $ar(G,\mathcal{T}_{q})=\ell_{q}(G)+1$, where $\ell_{q}(G)$ is the maximum size of a disconnected spanning subgraph $H$ of $G$ with the property that any two components of $H$ together have at most $q$ vertices. Using this equality, we obtain the exa…
A graph $F$ is Ramsey for a pair of graphs $(G,H)$ if any red/blue-coloring of the edges of $F$ yields a copy of $G$ with all edges colored red or a copy of $H$ with all edges colored blue. Two pairs of graphs are called Ramsey equivalent if they have the same collection of Ramsey graphs. The symmetric setting, that is, the case $G=H$, received considerable attention. This led to the open question whether there are connected graphs $G$ and $G'$ such that $(G,G)$ and $(G',G')$ are Ramsey equivalent. We make progress on the asymmetric version of this question and identify several non-trivial fam…
For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $Δ(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},$ $v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$, for $m\ge 7$ and $n\ge (m-3)^2+2$ we have $r(T_m^…
Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for $\ex(p;T_n^1)$ and $\ex(p;T_n^2)$, where $\ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,…
This article discusses some recent trends in Ramsey theory on infinite structures. Trees and their Ramsey theory have been vital to these investigations. The main ideas behind the author's recent method of trees with coding nodes are presented, showing how they can be useful both for coding structures with forbidden configurations as well as those with none. Using forcing as a tool for finite searches has allowed the development of Ramsey theory on such trees, leading to solutions for finite big Ramsey degrees of Henson graphs as well as infinite dimensional Ramsey theory of copies of the Rado…
The generalized Ramsey number $R(H, K)$ is the smallest positive integer $n$ such that for any graph $G$ with $n$ vertices either $G$ contains $H$ as a subgraph or its complement $\overline{G}$ contains $K$ as a subgraph. Let $T_n$ be a tree with $n$ vertices and $F_m$ be a fan with $2m + 1$ vertices consisting of $m$ triangles sharing a common vertex. We prove a conjecture of Zhang, Broersma and Chen for $m \ge 9$ that $R(T_n, F_m) = 2n - 1$ for all $n \ge m^2 - m + 1$. Zhang, Broersma and Chen showed that $R(S_n, F_m) \ge 2n$ for $n \le m^2 -m$ where $S_n$ is a star on $n$ vertices, implying…
We introduce the notion of (Ramsey) action of a tree on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary Hales--Jewett theorems (for both located and nonlocated words), and the Farah--Hindman--McLeod Ramsey theorem for layered actions on partial semigroups. We also establish a polynomial version of our main result, recovering the polynomial Milliken--Taylor theorem of Bergelson--Hindman--Williams as a particular case. We present applications of our Rams…
It was shown in \cite{sc12} that for a certain class of structures $\I$, $\I$-indexed indiscernible sets have the modeling property just in case the age of $\I$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we may conclude that certain classes of finite trees are Ramsey, some previously known. See updated paper for new references.
