Related papers: Ramsey numbers for bipartite graphs with small ban…

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\ldots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N,N}$ contains a…

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

An ordered graph $\mathcal{G}$ is a simple graph together with a total ordering on its vertices. The (2-color) Ramsey number of $\mathcal{G}$ is the smallest integer $N$ such that every 2-coloring of the edges of the complete ordered graph…

A seminal result of Lee asserts that the Ramsey number of any bipartite $d$-degenerate graph $H$ satisfies $\log r(H) = \log n + O(d)$. In particular, this bound applies to every bipartite graph of maximal degree $\Delta$. It remains a…

Let the star on $n$ vertices, namely $K_{1,n-1}$ be denoted by $S_n$. If every two coloring of the edges of a complete balanced multipartite graph $K_{j \times s}$ there is a copy of $S_n$ in the first color or a copy of $S_m$ in the second…

We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph $G$, denote the $t$-blowup of $G$ by $G[t]$. We say that $G$ is $r$-Ramsey for $H$, and write $G \stackrel{r}{\rightarrow} H$, if every…

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…

This paper begins by exploring some old and new results about Ramsey numbers and minimum numbers of monochromatic triangles in $2$-colorings of complete graphs, both in the disjoint and non-disjoint cases. We then extend the theory, by…

Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size of each vertex's list depends on its part. For complete bipartite graphs, we fix the list sizes of one part and consider the resulting…

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density…

For positive integers $n, m$, the double star $S(n,m)$ is the graph consisting of the disjoint union of two stars $K_{1,n}$ and $K_{1,m}$ together with an edge joining their centers. Finding monochromatic copies of double stars in…

Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.

The size-Ramsey number of a graph $F$ is the smallest number of edges in a graph $G$ with the Ramsey property for $F$, that is, with the property that any 2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove that…

Given two graphs H and G, the size multipartite Ramsey number mj (H, G) is the smallest natural number t such that an arbitrary coloring of the edges of Kjt, complete multipartite graph whose vertex set is partitioned into j parts each of…

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other:…

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we…

We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and…

In his study of graph codes, Alon introduced the concept of the odd-Ramsey number of a family of graphs $\mathcal{H}$ in $K_n$, defined as the minimum number of colours needed to colour the edges of $K_n$ so that every copy of a graph $H\in…

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or…

We say that $G$ is a $(3, 3)$-Ramsey graph if every $2$-coloring of the edges of $G$ forces a monochromatic triangle. The $(3, 3)$-Ramsey graph $G$ is minimal if $G$ does not contain a proper $(3, 3)$-Ramsey subgraph. In this work we find…