Related papers: Calculating Ramsey numbers by partitioning coloure…

In 2015 Bloom and Liebenau proved that $K_n$ and $K_n+K_{n-1}$ possess the same $2$-Ramsey graphs for all $n\geq 3$ (with a single exception for $n=3$). In the following we give a simple proof that $K_n$ and $K_n+K_{n-1}$ possess the same…

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$.…

R\"odl and Ruci\'nski (1990) established Ramsey's theorem for random graphs. In particular, for fixed integers $r$, $\ell\geq 2$ they showed that $\hat p_{K_\ell,r}(n)=n^{-\frac{2}{\ell+1}}$ is a threshold for the Ramsey property that every…

We prove that if $G$ is an $n$-vertex graph whose edges are coloured with red and blue, then the number of colour-alternating walks of length $2k+1$ with $k+1$ red edges and $k$ blue edges is at most $k^k(k+1)^{k+1}(2k+1)^{-2k-1}n^{2k+2}$.…

For $r:=(r_1,\dots,r_k)$, an $r$-factorization of the complete $\lambda$-fold $h$-uniform $n$-vertex hypergraph $\lambda K_n^h$ is a partition of (the edges of) $\lambda K_n^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is…

For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there…

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic…

A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses $k$ colors. Given an integer $k\ge1$ and graphs $H_1, \ldots, H_k$, the Gallai-Ramsey…

We prove that, for $n\geq 4$, the graphs $K_n$ and $K_n+K_{n-1}$ are Ramsey equivalent. That is, if $G$ is such that any red-blue colouring of its edges creates a monochromatic $K_n$ then it must also possess a monochromatic $K_n+K_{n-1}$.

P. Erd\H{o}s proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We…

We estimate the $3$-colour bipartite Ramsey number for balanced bipartite graphs $H$ with small bandwidth and bounded maximum degree. More precisely, we show that the minimum value of $N$ such that in any $3$-edge colouring of $K_{N,N}$…

The purpose is to study the strength of Ramsey's Theorem for pairs restricted to recursive assignments of $k$-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's…

We consider unavoidable chromatic patterns in $2$-colorings of the edges of the complete graph. Several such problems are explored being a junction point between Ramsey theory, extremal graph theory (Tur\'an type problems), zero-sum Ramsey…

Given an $r$-edge-colouring of the edges of a graph $G$, we say that it can be partitioned into $p$ monochromatic cycles when there exists a set of $p$ vertex-disjoint monochromatic cycles covering all the vertices of $G$. In the literature…

The reconfiguration graph $R_k(G)$ of the $k$-colourings of a graph~$G$ has as vertex set the set of all possible $k$-colourings of $G$ and two colourings are adjacent if they differ on exactly one vertex. We give a short proof of the…

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that…

The $k$-edge-colored bipartite Gallai-Ramsey number $\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that $n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors) of complete bipartite graph…

Let $G_1, G_2, ..., G_t$ be graphs. The multicolor Ramsey number $R(G_1, G_2, ..., G_t)$ is the smallest positive integer $n$ such that if the edges of complete graph $K_n$ are partitioned into $t$ disjoint color classes giving $t$ graphs…

Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. The question of finding exactly $m$-coloured complete subgraphs was first considered by Erickson…

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…