Related papers: A Probabilistic Threshold for Monochromatic Arithm…

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no $(k-\epsilon)^{\operatorname{pw}(G)}\operatorname{poly}(n)$ time algorithm for deciding if an $n$-vertex graph $G$ with pathwidth $\operatorname{pw}(G)$ admits a proper vertex…

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…

We study the minimum degree threshold $\delta_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluh\'ar and Treglown handled the 2-color case, proving that $\delta_{r,2} =…

Let D be a finite set of positive real numbers. The distance graph G(R,D) is the graph with vertex set R (set of real numbers), and two vertices x, y are adjacent if |x-y| belongs to D. We prove that every positive integer t>1 there is a…

A k-distance r-coloring of a graph is a coloring of the vertices of the graph such that if the distance between 2 vertices x and y is less or equal to k, then x and y must have distinct colors. A planar graph is a graph that can be drawn…

We examine the behavior of the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/n\mathbb{Z}$. We prove that if a set is chosen by including each element of $\mathbb{Z}/n\mathbb{Z}$ independently with constant…

We investigate the clustering transition undergone by an exemplary random constraint satisfaction problem, the bicoloring of $k$-uniform random hypergraphs, when its solutions are weighted non-uniformly, with a soft interaction between…

A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption…

This paper describes an improvement in the upper bound for the magnitude of a coefficient of a term in the chromatic polynomial of a general graph. If $a_r$ is the coefficient of the $q^r$ term in the chromatic polynomial $P(G,q)$, where…

We present a new short proof of Van der Waerden's Theorem about the existence of arbitrarily long monochromatic arithmetic progressions. The proof uses algebra in the compact space of ultrafilters $\beta\N$, but contrarily to the other…

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…

In this paper, we study discrepancy questions for spanning subgraphs of $k$-uniform hypergraphs. Our main result is that, for any integers $k \ge 3$ and $r \ge 2$, any $r$-colouring of the edges of a $k$-uniform $n$-vertex hypergraph $G$…

Liu, Pach and S\'andor recently characterized all polynomials $p(z)$ such that the equation $x+y=p(z)$ is $2$-Ramsey, that is, any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. In this paper,…

This paper introduces the Two-Class ($r$,$k$)-Coloring problem: Given a fixed number of $k$ colors, such that only $r$ of these $k$ colors allow conflicts, what is the minimal number of conflicts incurred by an optimal coloring of the…

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…

For many random Constraint Satisfaction Problems, by now, we have asymptotically tight estimates of the largest constraint density for which they have solutions. At the same time, all known polynomial-time algorithms for many of these…

Let $K_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a $2$-colouring of the edges of $K_{\mathbb{N}}$ in which every monochromatic path has density~$0$.…

Kosek, Robertson, Sabo, and Schaal studied the minimum number \(M_k(n)\) of monochromatic solutions to the strict Schur inequality system $x_1\le x_2\le x_3$ and $x_1+x_2<x_3$ in \(2\)-colorings of \([k+1,k+n]\). They proved that for every…