Related papers: Torus Queen Independence

The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp…

The n-dimensional hypercube Q_n is a graph with vertex set {0,1}^n such that there is an edge between two vertices if and only if they differ in exactly one coordinate. Let H be a graph, and a graph is called H-free if it does not contain H…

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the…

In 1959, Erd\H{o}s and Moser asked for the maximum number of unit distances that may be formed among the vertices of a convex $n$-gon; until now, the best known upper bound has been $2\pi n \log_2 n + O(n)$, achieved by F\"uredi in 1990. In…

We consider graphs on monomials in $n$ variables of a fixed degree $d$ where two monomials are adjacent if and only if their least common multiple has degree $d+1$. We prove that when $n = 3$ and $d$ is divisible by $3$ as well as when…

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-group, and $1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of…

The well-known three distance theorem states that there are at most three distinct gaps between consecutive elements in the set of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable…

The classical Steinitz theorem states that if the origin belongs to the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points of $S$ whose convex hull contains the origin in the interior.…

We study the disproportionate version of the classical cake-cutting problem: how efficiently can we divide a cake, here $[0,1]$, among $n$ agents with different demands $\alpha_1, \alpha_2, \dots, \alpha_n$ summing to $1$? When all the…

We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $\gamma\in (0,1)$ times its…

We revisit the problem of determining the independent domination number in hypercubes for which the known upper bound is still not tight for general dimensions. We present here a constructive method to build an independent dominating set…

Let $\mathcal{H}$ be a set of graphs. The planar Tur\'an number, $ex_\mathcal{P}(n,\mathcal{H})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{H}$ as a subgraph. When…

Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m…

For any real $x$ and any integer $k\ge1$, we say that a set $\mathcal{D}_{k}$ of $k$ distinct integers is a $k$-tuple jumping champion if it is the most common differences that occurs among $k+1$ consecutive primes less than or equal to…

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices…

A k-queue layout is a special type of a linear layout, in which the linear order avoids (k+1)-rainbows, i.e., k+1 independent edges that pairwise form a nested pair. The optimization goal is to determine the queue number of a graph, i.e.,…

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a…

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…