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The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is…

Given two graph families $\mathcal H_1$ and $\mathcal H_2$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder is trying to force Painter to…

We define the $r\textit{-Kneser Ramsey number}$ $R^{\textrm{KG}}_{r}(s, t)$ as the minimum integer $n$ such that every red/blue edge-coloring of the Kneser graph $\textrm{KG}(n,r)$ contains a red $s$-clique or a blue $t$-clique. We obtain…

Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in…

An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the…

A graph is properly edge-colored if no two adjacent edges have the same color. The smallest number of edges in a graph any of whose proper edge colorings contains a totally multicolored copy of a graph H is the size anti-Ramsey number…

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 $\overline{R}(\mathcal{H},\mathcal{G})$…

Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every $\{$red, blue$\}$-coloring of the edges of $G$ contains a red copy of $H_1$ or a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if $G…

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…

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (H)_r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G…

Given bipartite graphs $G_1, \ldots, G_n$, the bipartite Ramsey number $BR(G_1,\ldots, G_n)$ is the last integer $b$ such that any complete bipartite graph $K_{b,b}$ with edges coloured with colours $1,2,\ldots,n$ contains a copy of some…

For two given graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the smallest integer $N$ such that any red-blue edge-coloring of the complete graph $K_N$ contains a red $F$ or a blue $H$. When $F=H$, we simply write $R_2(H)$. For an…

For given graphs $G_{1}, G_{2}, ... , G_{k}, k \geq 2$, the multicolor Ramsey number $R(G_{1}, G_{2}, ... , G_{k})$ is the smallest integer $n$ such that if we arbitrarily color the edges of the complete graph of order $n$ with $k$ colors,…

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} $RT(n,H,\rho-mean)$…

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic…

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…

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set…

We extend two well-known results in Ramsey theory from from $K_n$ to arbitrary $n$-chromatic graphs. The first is a note of Erd\H os and Rado stating that in every 2-coloring of the edges of $K_n$ there is a monochromatic tree on $n$…

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 $G$, 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 $D$.…