Related papers: Tur\'an's problem and generalized Ramsey numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all…

The Ramsey number $R(C_4,K_m)$ is the smallest $n$ such that any graph on $n$ vertices contains a cycle of length four or an independent set of order $m$. With the help of computer algorithms we obtain the exact values of the Ramsey numbers…

We prove a generalised Ramsey--Tur\'an theorem for matchings, which (a) simultaneously generalises the Cockayne--Lorimer Theorem (Ramsey for matchings) and the Erd\H{o}s--Gallai Theorem (Tur\'an for matchings), and (b) is a generalised…

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…

A $(p,q)$-coloring of a graph $G$ is an edge-coloring of $G$ such that every $p$-clique receives at least $q$ colors. In 1975, Erd\H{o}s and Shelah introduced the generalized Ramsey number $f(n,p,q)$ which is the minimum number of colors…

Let $K\_{[k,t]}$ be the complete graph on $k$ vertices from which a set of edges, induced by a clique of order $t$, has been dropped. In this note we give two explicit upper bounds for $R(K\_{[k\_1,t\_1]},\dots, K\_{[k\_r,t\_r]})$ (the…

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,…

Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic copy of $H$. The existence of these numbers has been known since 1930 but their…

Given a positive integer $s$, a graph $G$ is $s$-Ramsey for a graph $H$, denoted $G\rightarrow (H)_s$, if every $s$-colouring of the edges of $G$ contains a monochromatic copy of $H$. The $s$-colour size-Ramsey number ${\hat{r}}_s(H)$ of a…

We study Ramsey-type problems in Gallai-colorings. Given a graph $G$ and an integer $k\ge1$, the Gallai-Ramsey number $gr_k(K_3,G)$ is the least positive integer $n$ such that every $k$-coloring of the edges of the complete graph on $n$…

Assume that $K_{j\times n}$ be a complete, multipartite graph consisting of $j$ partite sets and $n$ vertices in each partite set. For given graphs $G_1, G_2,\ldots, G_n$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2,…

Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its…

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 $E=\{v_0v_1,\ldots,v_0v_{n-3},$…

Let $G$, $H$ and $K$ represent three graphs without loops or parallel edges and $n$ represent an integer. Given any red blue coloring of the edges of $G$, we say that $K \rightarrow (G,H)$, if there exists red copy of $G$ in $K$ or a blue…

Assume that $K_{j\times n}$ be a complete, multipartite graph consisting of $j$ partite sets and $n$ vertices in each partite set. For given graphs $G_1$ and $G_2$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2)$ is the smallest…

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1 \leq s \leq r-1$, an $s$-path ${\mathcal P}^{r,s}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\ldots,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\ldots,…

Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+ $, the Ramsey-Tur\'an number of $H$ and $f$, denoted by $RT(n, H, f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not…

Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\{1,2, \dots , n \}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of…

For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting…

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a…