Related papers: Hypergraph Ramsey numbers of cliques versus stars

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform…

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…

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show…

Burr and Erd\H{o}s in 1975 conjectured, and Chv\'atal, R\"odl, Szemer\'edi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed…

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set…

The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{\H o}s and…

We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number $\overline{R}_e(\mathfrak{G})$ of an edge-ordered graph $\mathfrak{G}$ is the…

An $r$-uniform hypergraph $H$ is semi-algebraic of complexity $\mathbf{t}=(d,D,m)$ if the vertices of $H$ correspond to points in $\mathbb{R}^{d}$, and the edges of $H$ are determined by the sign-pattern of $m$ degree-$D$ polynomials.…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…

The grid Ramsey number $ G(r) $ is the smallest number $ n $ such that every edge-colouring of the grid graph $\Gamma_{n,n} := K_n \times K_n$ with $r$ colours induces a rectangle whose parallel edges receive the same colour. We show $ G(r)…

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…

For graphs $F$ and $G$, let $F\to G$ signify that any red/blue edge coloring of $F$ contains a monochromatic $G$. Denote by ${\cal G}(N,p)$ the random graph space of order $N$ and edge probability $p$. Using the regularity method, one can…

A uniform hypergraph $H$ is called $k$-Ramsey for a hypergraph $F$, if no matter how one colors the edges of $H$ with $k$ colors, there is always a monochromatic copy of $F$. We say that $H$ is minimal $k$-Ramsey for $F$, if $H$ is…

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…

For a positive integer $r$, let $G(r)$ be the smallest $N$ such that, whenever the edges of the Cartesian product $K_N \times K_N$ are $r$-coloured, then there is a rectangle in which both pairs of opposite edges receive the same colour. In…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c…

For a graph $H$ and an integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote the $k$-color Ramsey number and list Ramsey number of $H$, respectively. Alon, Buci\'c, Kalvari, Kuperwasser and Szab\'o in 2021 initiated the systematic study of…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

An ordered graph $H$ on $n$ vertices is a graph whose vertices have been labeled bijectively with $\{1,...,n\}$. The ordered Ramsey number $r_<(H)$ is the minimum $n$ such that every two-coloring of the edges of the complete graph $K_n$…