Related papers: Recent Progress in Ramsey Theory

For a graph $L$ and an integer $k\geq 2$, $R_k(L)$ denotes the smallest integer $N$ for which for any edge-colouring of the complete graph $K_N$ by $k$ colours there exists a colour $i$ for which the corresponding colour class contains $L$…

Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$ or a blue copy of $H_2$. In this paper we investigate Ramsey questions in the setting of…

The Ramsey number is the minimum number of nodes, $n = R(s, t)$, such that all undirected simple graphs of order $n$, contain a clique of order $s$, or an independent set of order $t$. This paper explores the application of a best first…

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform…

Let $G$ be a graph, $H$ be a subgraph of $G$, and let $G- H$ be the graph obtained from $G$ by removing a copy of $H$. Let $K_{1, n}$ be the star on $n+ 1$ vertices. Let $t\geq 2$ be an integer and $H_{1}, \dots, H_{t}$ and $H$ be graphs,…

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…

A $\rho$-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most $\rho$. For a graph $H$ and for $\rho \geq 1$, the {\em mean Ramsey-Tur\'an number} $RT(n,H,\rho-mean)$…

Motivated by an extremal problem on graph-codes that links coding theory and graph theory, Alon recently proposed a question aiming to find the smallest number $t$ such that there is an edge coloring of $K_{n}$ by $t$ colors with no copy of…

Let $G$ and $H$ be finite graphs. If for any two-coloring of the edges of a complete graph $K_n$, there is a copy of $G$ in the first color, red, or a copy of $H$ in the second color, blue, we will say $K_n\rightarrow (G,H)$. The Ramsey…

An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the…

In 2011, Meunier conjectured that for positive integers $n,k,r,s$ with $ k\geq 2$, $r\geq 2$, and $n\geq \max (\{r,s\})k$, the chromatic number of $s$ -stable $r$-uniform Kneser hypergraphs is equal to $\left\lceil \frac{n-\max…

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H and H' are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H'. In this paper, we study…

In Ramsey theory for graphs we are given a graph $G$ and we are required to find the least $n_0$ such that, for any $n\geq n_0$, any red/blue colouring of the edges of $K_n$ gives a subgraph $G$ all of whose edges are blue or all are red.…

A graph $G$ is said to be $q$-Ramsey for a $q$-tuple of graphs $(H_1,\ldots,H_q)$, denoted by $G\to_q(H_1,\ldots,H_q)$, if every $q$-edge-coloring of $G$ contains a monochromatic copy of $H_i$ in color $i,$ for some $i\in[q]$. Let…

We study the structure of red-blue edge colorings of complete graphs, with no copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n \ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any…

Ramsey-Tur\'{a}n type problems were initiated by Erd\H{o}s and S\'{o}s in 1969. Given integers $p, q\ge2$, a graph $G$ is $(K_p,K_q)$-free if there exists a red/blue edge coloring of $G$ such that it contains neither a red $K_p$ nor a blue…

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in…

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$…

Let $G, H$ be finite graphs without loops or multiple edges and $K_n$ denote the complete graph on $n$ vertices. If for every red/blue colouring of edges of the complete graph $K_n$, there exists a red copy of $G$, or a blue copy of $H$, we…

Let $R(k_1, \cdots, k_r)$ denote the classical $r$-color Ramsey number for integers $k_i \ge 2$. The Diagonal Conjecture (DC) for classical Ramsey numbers poses that if $k_1, \cdots, k_r$ are integers no smaller than 3 and $k_{r-1} \leq…