Related papers: Bounds on Arithmetic Rainbow Ramsey Multiplicities

The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted $r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $G$. The…

Let $G$ be an edge colored graph. A {\it}{rainbow path} in $G$ is a path in which all the edges are colored with distinct colors. Let $d^c(v)$ be the color degree of a vertex $v$ in $G$, i.e. the number of distinct colors present on the…

In 2014, Moshkovitz and Shapira determined the tower height for hypergraph Ramsey numbers of tight monotone paths. We address the color-avoiding version of this problem in which one no longer necessarily seeks a monochromatic subgraph, but…

For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$…

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic…

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one…

Mantel's Theorem asserts that a simple $n$ vertex graph with more than $\frac{1}{4}n^2$ edges has a triangle (three mutually adjacent vertices). Here we consider a rainbow variant of this problem. We prove that whenever $G_1, G_2, G_3$ are…

Given two graphs $G$ and $H$, the {\it rainbow number} $rb(G,H)$ for $H$ with respect to $G$ is defined as the minimum number $k$ such that any $k$-edge-coloring of $G$ contains a rainbow $H$, i.e., a copy of $H$, all of its edges have…

We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number…

We study edge-colorings of the complete $p$-graph on $n$ vertices that contain no three edges $A,B,C$ of distinct colors such that the symmetric difference of $A$ and $B$ is contained in $C$. For $p\ge3$ and $n\ge p+1$, we show that every…

Jungi\'{c} et al (2003) defined $T_{k}$ as the minimal number $t \in \mathbb{N}$ such that there is a rainbow arithmetic progression of length $k$ in every equinumerous $t$-coloring of $[t n]$ for every $n \in \mathbb{N}$. They proved that…

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number of copies of $H$ that a properly edge-coloured graph on $n$…

An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $ar_r(n,\mathcal{G})$ is…

In the paper we state and prove theorem describing the upper bound on number of the graphs that have fixed number of vertices |V| and can be colored with the fixed number of n colors. The bound relates both numbers using power of 2, while…

The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd\H{o}s…

Given two graphs $G$ and $H$, let $f(G,H)$ denote the maximum number $c$ for which there is a way to color the edges of $G$ with $c$ colors such that every subgraph $H$ of $G$ has at least two edges of the same color. Equivalently, any…

The rainbow Tur\'an number, a natural extension of the well studied traditional Tur\'an number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstra\"ete. The rainbow Tur\'an number of a graph $H$, $ex^{*}(n,H)$, is the largest…

Let $F$ be a fixed graph. The rainbow Tur\'an number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (where a rainbow copy of $F$ means a copy of $F$…

In this paper we introduce a general framework for proving lower bounds for various Ramsey type problems within random settings. The main idea is to view the problem from an algorithmic perspective: we aim at providing an algorithm that…

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,n\geq 3$, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey…