Related papers: Complementary Ramsey numbers and Ramsey graphs

In this paper we define new numbers called the Neo-Ramsay numbers. We show that these numbers are in fact equal to the Ramsay numbers. Neo-Ramsey numbers are easy to compute and for finding them it is not necessary to check all possible…

For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or…

The Ramsey number $R(s,t)$ is the smallest integer $n$ such that all graphs of size $n$ contain a clique of size $s$ or an independent set of size $t$. $\mathcal{R}(s,t,n)$ is the set of all counterexample graphs without this property for a…

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of…

The purpose of this survey is to provide a gentle introduction to several recent breakthroughs in graph Ramsey theory. In particular, we will outline the proofs (due to various groups of authors) of exponential improvements to the diagonal,…

The Ramsey number for the pair of graphs $\mathbb{K}_{1,n}$ (star) versus $W_{m}$ (wheel) has been extensively studied. In contrast, the Ramsey number of $\mathbb{K}_{2,n}$ versus the wheel is not yet explored due to the bit more structural…

For simple graphs $G$ and $H$, their size Ramsey number $\hat{r}(G,H)$ is the smallest possible size of $F$ such that for any red-blue coloring of its edges, $F$ contains either a red $G$ or a blue $H$. Similarly, we can define the…

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,n\geq 3$, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey…

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$…

The Ramsey number $R(s,t)$ is the least integer $n$ such that any coloring of the edges of $K_n$ with two colors produces either a monochromatic $K_s$ in one color or a monochromatic $K_t$ in the other. If $s=t$, we say that the Ramsey…

We define and develop preliminary theoretical results for the $\Gamma$-switch Ramsey number, a variation on the classical $m$-colour Ramsey number for which we allow permuting the colours incident with a vertex using elements of a group…

We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically…

Operator systems of matrices can be viewed as quantum analogues of finite graphs. This analogy suggests many natural combinatorial questions in linear algebra. We determine the quantum Ramsey numbers $QR(2,k)$ and the lower quantum Tur\'an…

In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense and $k$-sparse sets, both commonly called $k$-defective sets. We focus on the…

The two-colour Ramsey number $R(m,n)$ is the least natural number $p$ such that any graph of order $p$ must contain either a clique of size $m$ or an independent set of size $n$. We exhibit a method for computing upper bounds for $R(m,n)$…

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small…

We obtain some new upper bounds on the Ramsey numbers of the form $R(\underbrace{C_4,\ldots,C_4}_m,G_1,\ldots,G_n)$, where $m\ge 1$ and $G_1,\ldots,G_n$ are arbitrary graphs. We focus on the cases of $G_i$'s being complete, star $K_{1,k}$…

Let $P_t$ denote the path on $t$ vertices. The $r$-coloured Ramsey number of $P_t$, denoted by $R_r(P_t)$, is the minimum integer $n$ such that whenever the complete graph on $n$ vertices is given an $r$-edge-colouring, there exists a…

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a…

For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\Delta(T_n')=n-2$, and let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},$…