Related papers: The Ramsey numbers of paths versus wheels: a compl…

For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $…

For graphs $G_1,\ldots,G_k$, the Ramsey number $R(G_1,\ldots,G_k)$ is the smallest positive integer $N$ such that every $k$-edge-coloring of $K_N$ contains a monochromatic copy of $G_i$ in color $i$ for some $i\in[k]$. The Gallai--Ramsey…

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ on such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. Motivated…

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$,…

The Ramsey number $r(G)$ of a graph $G$ is the minimum number $N$ such that any red-blue colouring of the edges of $K_N$ contains a monochromatic copy of $G$. Pavez-Sign\'e, Piga and Sanhueza-Matamala proved that for any function $n\leq…

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly…

The weighted Ramsey number, ${\rm wR}(n,k)$, is the minimum $q$ such that there is an assignment of nonnegative real numbers (weights) to the edges of $K_n$ with the total sum of the weights equal to ${n\choose 2}$ and there is a Red/Blue…

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma…

For graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the \ordered Ramsey number $r_<(G^<,H^<)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete ordered graph $K^<_N$ on $N$ vertices…

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,…

The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd\H{o}s…

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2},…

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

A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored the same. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to…

The anti-Ramsey problem was introduced by Erd\H{o}s, Simonovits and S\'{o}s in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the…

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the Gallai-Ramsey…

A celebrated result of Chv\'atal, R\"odl, Szemer\'edi and Trotter states (in slightly weakened form) that, for every natural number $\Delta$, there is a constant $r_\Delta$ such that, for any connected $n$-vertex graph $G$ with maximum…

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…

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$ contains either a rainbow…

The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges…