Related papers: New lower bounds for van der Waerden numbers

An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gy\'arf\'as, P\'alv\"olgyi, Patk\'os and Wales proved that there exists a number $g(k)$ such that $n\geq g(k)$ if and only if…

The $p$-partite Ramsey number for quadrilateral, denoted by $r_p(C_4,k)$, is the least positive integer $n$ such that any coloring of the edges of a complete $p$-partite graph with $n$ vertices in each partition with $k$ colors will result…

Let $k$ and $n$ be positive integers, $n>k$. Define $r(n,k)$ to be the minimum positive value of $$ |\sqrt{a_1} + ... + \sqrt{a_k} - \sqrt{b_1} - >... -\sqrt{b_k} | $$ where $ a_1, a_2, ..., a_k, b_1, b_2, ..., b_k $ are positive integers…

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems. 3-SAT is equivalent to…

An acyclic edge coloring of a graph is a proper edge coloring without any bichromatic cycles. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum $k$ such that $G$ has an acyclic edge coloring with $k$ colors.…

We investigate the classical and distributed complexity of \emph{$k$-partial $c$-coloring} where $c=k$, a natural generalization of Brooks' theorem where each vertex should be colored from the palette $\{1,\ldots,c\} = \{1,\ldots,k\}$ such…

We study problems related to colouring bottomless rectangles. One of our main results shows that for any positive integers $m, k$, there is no semi-online algorithm that can $k$-colour bottomless rectangles with disjoint boundaries in…

We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any…

We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$ minimizes the number of monochromatic arithmetic 3-progressions under certain restrictions. A monochromatic arithmetic progression is a set of equally-spaced integers…

A weak $c$-colouring of a balanced incomplete block design (BIBD) is a colouring of the points of the design with $c$ colours in such a way that no block of the design has all of its vertices receive the same colour. A BIBD is said to be…

Bounded expansion and nowhere-dense classes of graphs capture the theoretical tractability for several important algorithmic problems. These classes of graphs can be characterized by the so-called weak coloring numbers of graphs, which…

We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erd\H{o}s and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2…

Given a graph $G$, a proper $k$-coloring of $G$ is a partition $c = (S_i)_{i\in [1,k]}$ of $V(G)$ into $k$ stable sets $S_1,\ldots, S_{k}$. Given a weight function $w: V(G) \to \mathbb{R}^+$, the weight of a color $S_i$ is defined as $w(i)…

We introduce a decoder for the 3D color code with boundaries, which is a variation of the restriction decoder introduced by Kubicka and Delfosse. Specifically, we adapt the lift procedure to efficiently find a correction on qubits adjacent…

For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to $\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For…

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…

The generalized Ramsey number $r(G, H, q)$ is the minimum number of colors needed to color the edges of $G$ such that every isomorphic copy of $H$ has at least $q$ colors. In this note, we improve the upper and lower bounds on $r(K_{n, n},…

In the $\ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $\ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an…

We introduce the van der Waerden complex ${\rm vdW}(n,k)$ defined as the simplicial complex whose facets correspond to arithmetic progressions of length $k$ in the vertex set $\{1, 2, \ldots, n\}$. We show the van der Waerden complex ${\rm…

We prove that if $A\subseteq \{1,\dots,N\}$ does not contain any non-trivial three-term arithmetic progression, then $$|A|\ll \frac{(\log\log N)^{3+o(1)}}{\log N}N\,.$$