Related papers: Vector space Ramsey numbers and weakly Sidorenko a…

An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an…

We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq…

In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a…

The fourth listed author and Hans Parshall (\cite{IosevichParshall}) proved that if $E \subset {\mathbb F}_q^d$, $d \ge 2$, and $G$ is a connected graph on $k+1$ vertices such that the largest degree of any vertex is $m$, then if $|E| \ge C…

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…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The feasible region $\Omega(\mathcal{F})$ of $\mathcal{F}$ is the set of points $(x,y)$ in the unit square such that there exists a sequence of $\mathcal{F}$-free $r$-uniform…

We use $p$-rank bounds on partial ovoids and the classical bounds on Ramsey numbers to obtain upper bounds on the size of partial $m$-ovoids in finite classical polar spaces. These bounds imply non-existence of $m$-ovoids for new infinite…

For two graphs $G_1$ and $G_2$, the size Ramsey number $\hat{r}(G_1,G_2)$ is the smallest positive integer $m$ for which there exists a graph $G$ of size $m$ such that for any red-blue edge-coloring of the graph $G$, $G$ contains either a…

We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\ge \rho q^2$, with $\frac{1}{\sqrt{q}}<<\rho \leq 1$, then it contains an isometric copy of $\ge c\rho q^3$ triangles.

Building on recent work of Mattheus and Verstra\"ete, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an…

The multicolor Ramsey number $r_k(F)$ of a graph $F$ is the least integer $n$ such that in every coloring of the edges of $K_n$ by $k$ colors there is a monochromatic copy of $F$. In this short note we prove an upper bound on $r_k(F)$ for a…

The Ramsey number $r(t;\ell)$ is the smallest $n$ such that every $\ell$-coloring of the edges of $K_n$ gives a monochromatic $K_{t}$. In recent years, there have been several improvements on asymptotic lower bounds for these numbers when…

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…

For graphs $F$, $G$, and $H$, we write $F \to (G,H)$ if every red-blue coloring of the edges of $F$ produces a red copy of $G$ or a blue copy of $H$. The graph $F$ is said to be $(G,H)$-minimal if it is subgraph-minimal with respect to this…

For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was…

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…

We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided…

Ramsey's theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this…

For graphs $G$, $F$ and $H$, let $G\rightarrow (F,H)$ signify that any edge coloring of $G$ in red and blue contains a red $F$ or a blue $H$. The Ramsey number $R(F,H)=\min\{r|\; K_r\rightarrow (F,H)\}$. In this note, we consider redundant…

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) =…