Related papers: Bounded VC-dimension implies the Schur-Erdos conje…

The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone…

The \emph{book graph} of order $(n+2)$, denoted by $B_{n}$, is the graph with $n$ distinct copies of triangles sharing a common edge called the `base'. A cycle of order $m$ is denoted by $C_{m}$. A lot of studies have been done in recent…

For graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. A book $B_n$ is a graph consisting of $n$…

We prove an old conjecture of Erd{\H o}s and Graham on sums of unit fractions: There exists a constant $b>0$ such that if we $r$-color the integers in $2,b^r]$, then there exists a monochromatic set $S$ such that $\sum_{n \in S} 1/n=1$.

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $…

A set of vertices $X\subseteq V$ in a simple graph $G(V,E)$ is irredundant (CO-irredundant) if each vertex $x\in X$ is either isolated in the induced subgraph $G[X]$ or else has a private neighbor $y\in V\setminus X$ ($y\in V$) that is…

In 1978, Chung and Liu generalized the definition of the Ramsey number. They introduced the $d$-chromatic Ramsey number as follows. Let $1\leq d< c$ be integers and let $A_{1}, \dots, A_{t}$ be subsets with size $d$ of $[c]$, where $t=…

We investigate the Ramsey numbers $r(I_m, L_n)$ which is the minimal natural number $k$ such that every oriented graph on $k$ vertices contains either an independent set of size $m$ or a transitive tournament on $n$ vertices. Apart from the…

The square $G^2$ of a graph $G$ is the graph on $V(G)$ with a pair of vertices $uv$ an edge whenever $u$ and $v$ have distance $1$ or $2$ in $G$. Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the…

The odd-Ramsey number $r_{\text{odd}}(n,H)$ of a graph $H$ is the minimum number of colors needed to edge-color $K_n$ so that in every copy of $H$ some color occurs an odd number of times, and the unique-Ramsey number $r_{\text{u}}(n,H)$ is…

For a positive integer $k$ and a graph $H$, the $k$-color induced size-Ramsey number $\hat{R}_{\mathrm{ind}}(H, k)$ is the minimum integer $m$ for which there exists a graph $G$ with $m$ edges such that for every $k$-edge coloring of $G$,…

Fix integers $d,r\ge 2$ and suppose that the edge set of the $d$-fold Cartesian product of the $N$-clique $K_N^d$ is $r$-colored. We show that there is a copy of $K_n^d$ whose edges in each direction are monochromatic provided $N > 2^{2^{c…

The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $n$ for which every $k$-edge-coloured complete bipartite graph $K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was…

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This…

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$ (that is, it consists of a matching and some isolated vertices) and the…

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 $…

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey…

We study two problems in graph Ramsey theory. In the early 1970's, Erd\H{o}s and O'Neil considered a generalization of Ramsey numbers. Given integers $n,k,s$ and $t$ with $n \ge k \ge s,t \ge 2$, they asked for the least integer…

In this simple paper, we exhibit a Schur partition giving rise to a triangle-free linear colouring of $K_{1697}$ in 7 colours. Thus we show that the Schur number $S(7) \ge 1696$ and the multicolour Ramsey number $R_{7}(3) \ge 1698$. We also…

Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey…