Related papers: Cardinalities of $g$-difference sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical…

We show that for any $t>1$, the set of unconditional convex bodies in $\mathbb{R}^n$ contains a $t$-separated subset of cardinality at least $\exp \exp (C(t) n)$. This implies that there exists an unconditional convex body in $\mathbb{R}^n$…

In this paper, we consider the problem of finding a maximum cardinality subset of vectors, given a constraint on the normalized squared length of vectors sum. This problem is closely related to Problem 1 from (Eremeev, Kel'manov, Pyatkin,…

Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We…

In this paper we rectify two previous results found in the literature. Our work leads to a new upper bound for the largest sum-free subset of $[1,n]$ with lowest value in $\left [\frac{n}{3},\frac{n}{2}\right ]$, and the identification of…

Reed conjectured that for every $\varepsilon>0$ and every integer $\Delta$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\Delta$ and girth at least $g$ is at most…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane…

Let ||.|| be a norm in R^d whose unit ball is B. Assume that V\subset B is a finite set of cardinality n, with \sum_{v \in V} v=0. We show that for every integer k with 0 \le k \le n, there exists a subset U of V consisting of k elements…

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq…

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient $F$, over any arithmetic progression. Namely, if $g_{a,b}(x)$ denotes the number of even…

Let $G$ be a finite permutation group acting on a set $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer of the sequence is trivial and no point…

Let $\alpha(G)$ denote the cardinality of a maximum independent set. An independent set $I$ of $G$ is critical if $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$ for every independent set $J$ of $G$. Let…

An end of a graph $G$ is an equivalence class of rays, where two rays are equivalent if there are infinitely many vertex-disjoint paths between them in $G$. The degree of an end is the maximum cardinality of a collection of pairwise…

We give a definition of some classes of boolean algebras generalizing free boolean algebras; they satisfy a universal property that certain functions extend to homomorphisms. We give a combinatorial property of generating sets of these…

We study the local properties problem for difference sets: If we have a large set of real numbers and know that every small subset has many distinct differences, to what extent must the entire set have many distinct differences? More…

Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the distance between $u$ and $w$ does not equal the distance between $v$ and $w$, then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted $\beta(G)$,…

For every $n\in\mathbb N$ we construct a finite graph $G$ such that every orientation $\vec G$ of $G$ contains an isometric copy of any oriented tree on $n$ vertices, and evaluate the smallest possible cardinality of $G$. On the other hand,…

We show that the maximum cardinality of an anti-chain composed of intersections of a given set of n points in the plane with half-planes is close to quadratic in n. We approach this problem by establishing the equivalence with the problem…