Related papers: Ramsey numbers and the Zarankiewicz problem

The Zarankiewicz problem asks for an estimate on $z(m, n; s, t)$, the largest number of $1$'s in an $m \times n$ matrix with all entries $0$ or $1$ containing no $s \times t$ submatrix consisting entirely of $1$'s. We show that a classical…

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 $K_{1,k}$…

For positive integers $s$, $t$, $m$ and $n$, the Zarankiewicz number $Z_{s,t}(m,n)$ is defined to be the maximum number of edges in a bipartite graph with parts of sizes $m$ and $n$ that has no complete biparitite subgraph containing $s$…

Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and…

For positive integers $s,t,m$ and $n$, the Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a subgraph of $K_{m,n}$ that has no complete bipartite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices…

Given positive integers m,n,s,t, let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m-by-n that does not contain an all ones submatrix of size s-by-t. We find a flexible upper bound on z(m,n,s,t) that implies the known…

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by…

For any positive integers $k$ and $n$, let $B_n^{(k)}$ be the book graph consisting of $n$ copies of the complete graph $K_{k+1}$ sharing a common $K_k$. Let $C_m$ be a cycle of length $m$. Prior work by Allen, \L uczak, Polcyn, and Zhang…

Let $n\geq\nu$, let $T$ be an $n$-vertex tree with bipartition class sizes $t_1\geq t_2$, and let $S$ be a $\nu$-vertex tree with bipartition class sizes $\tau_1\geq\tau_2$. Using four natural constructions, we show that the Ramsey number…

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most…

The `global' Zarankiewicz problem for hypergraphs asks for an upper bound on the number of edges of a finite $r$-hypergraph $V$ in terms of the number $|V|$ of its vertices, assuming the edge relation is induced by a fixed $K_{k, \dots,…

The Ramsey numbers $R(T_n,W_8)$ are determined for each tree graph $T_n$ of order $n\geq 7$ and maximum degree $\Delta(T_n)$ equal to either $n-4$ or $n-5$. These numbers indicate strong support for the conjecture, due to Chen, Zhang and…

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 wheel $W_{k}$ is the graph on $k+1$ vertices consisting of a vertex joined to a cycle of length $k$, and we say that $W_k$ is an even wheel if $k$ is even. Mao, Wang, Magnant, Schiermeyer proved that the Ramsey number of $W_{2n}$ is…

Let $R(H_1,H_2)$ denote the Ramsey number for the graphs $H_1, H_2$, and let $J_k$ be $K_k{-}e$. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several…

An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More…

The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete…

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is…

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…

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique…