Related papers: Improved Extractors for Affine Lines

Given a graph $H$, the size Ramsey number $r_e(H,q)$ is the minimal number $m$ for which there is a graph $G$ with $m$ edges such that every $q$-coloring of $G$ contains a monochromatic copy of $H$. We study the size Ramsey number of the…

W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left\{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right\} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace…

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring $\varphi$ of the $d$-dimensional hypercube $Q_d$, we are interested in whether there is a proper $d$-edge…

We investigate proper $(a:b)$-fractional colorings of $n$-uniform hypergraphs, which generalize traditional integer colorings of graphs. Each vertex is assigned $b$ distinct colors from a set of $a$ colors, and an edge is properly colored…

In this paper we introduce and study a new problem named \emph{min-max edge $q$-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer $q$. The goal is…

A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is a proper coloring in which every color class is contained in the neighbourhood of some vertex. The minimum number of colors required for any…

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer…

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits,…

Using the framework of the Chromo Dielectric Model we perform an analysis of color electric flux tubes in meson-like $q\bar{q}$ and baryon-like $qqq$ quark configurations. We discuss the Abelian color structure of the model and point out a…

There are many extremely challenging problems about existence of monochromatic arithmetic progressions in colorings of groups. Many theorems hold only for abelian groups as results on non-abelian groups are often much more difficult to…

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points at distance one. Answering a question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is a natural number $m$ and a…

For an integer $q\ge 2$ and an even integer $d$, consider the graph obtained from a large complete $q$-ary tree by connecting with an edge any two vertices at distance exactly $d$ in the tree. This graph has clique number $q+1$, and the…

In Euclidean Ramsey Theory usually we are looking for monochromatic configurations in the Euclidean space, whose points are colored with a fixed number of colors. In the canonical version, the number of colors is arbitrary, and we are…

The study of upper density problems on Ramsey theory was initiated by Erd\H{o}s and Galvin in 1993. In this paper we are concerned with the following problem: given a fixed finite graph $F$, what is the largest value of $\lambda$ such that…

Given a so called ''Sperner coloring'' of a triangulation of the $D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all $D+1$ colors. However, finding a rainbow simplex was the…

Given a graph, an edge coloring assigns colors to edges so that no pairs of adjacent edges share the same color. We are interested in edge coloring algorithms under the W-streaming model. In this model, the algorithm does not have enough…

For the Hamming graph $H(n,q)$, where a $q$ is a constant prime power and $n$ grows, we construct perfect colorings without non-essential arguments such that $n$ depends exponentially on the off-diagonal part of the quotient matrix. In…

Let $r,s,t\geq2$ be integers. For $r$-graphs $G$ and $F_1,\dots,F_s$, we write $G\to(F_1,\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\leq i\leq s$. Let…

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…