Related papers: Ramsey problems for monotone paths in graphs and h…
Given a graph $H$ and a positive integer $k$, the {\it $k$-colored Ramsey number} $R_k(H)$ is the minimum integer $n$ such that in every $k$-edge-coloring of the complete graph $K_{n}$, there is a monochromatic copy of $H$. Given two graphs…
The Ramsey number r(H) of a graph H is the minimum positive integer N such that every two-coloring of the edges of the complete graph K_N on N vertices contains a monochromatic copy of H. A graph H is d-degenerate if every subgraph of H has…
The classical Ramsey numbers $r(s,t)$ denote the minimum $n$ such that every red-blue coloring of the edges of the complete graph $K_n$ contains either a red clique of order $s$ or a blue clique of order $t$. These quantities are the…
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding…
In this short note we prove that there is a constant $c$ such that every k-edge-coloring of the complete graph K_n with n > 2^{ck} contains a K_4 whose edges receive at most two colors. This improves on a result of Kostochka and Mubayi, and…
In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph…
We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey…
A graph $G$ is said to be Ramsey for a tuple of graphs $(H_1,\dots,H_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$, for some $i$. A fundamental question at the intersection of Ramsey…
The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey…
Let $k \in \mathbb{N}$ and let $H_1, H_2, \ldots, H_k$ be simple graphs such that for each $j \in \{ 1, 2, \ldots, k \}$, the vertex set of $H_j$ is $\{ 0, 1, 2, \ldots, n_j - 1 \}$ for some $n_j \in \mathbb{N}$. The ordered Ramsey number…
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the…
Ramsey's theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey's theorem for pairs due to Erdos and Rado, which states that every…
An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…
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…
A connected matching in a graph $G$ is a matching contained in a connected component of $G$. A well-known method due to {\L}uczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic…
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…
We determine the anti-Ramsey numbers for paths. This confirms a conjecture posed by Erd\H{o}s, Simonovits and S\'{o}s in 1970s.
The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one…
An ordered graph is a graph with a linear ordering on its vertices. The online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite sequence of vertices; on each turn, Builder draws an edge between two vertices, and Painter…
For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there…