Related papers: A New Upper Bound for Cancellative Pairs

A finite family ${\mathcal B}$ of balls with respect to an arbitrary norm in ${\mathbb R}^d$ ($d\geq 2$) is called a non-separable family if there is no hyperplane disjoint from $\bigcup {\mathcal B}$ that strictly separates some elements…

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show…

In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{…

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by…

The deletion distance between two binary words $u,v \in \{0,1\}^n$ is the smallest $k$ such that $u$ and $v$ share a common subsequence of length $n-k$. A set $C$ of binary words of length $n$ is called a $k$-deletion code if every pair of…

Let $\mbox{$\cal F$}\subseteq 2^{[n]}$ be a fixed family of subsets. Let $D(\mbox{$\cal F$})$ stand for the following set of Hamming distances: $$ D(\mbox{$\cal F$}):=\{d_H(F,G):~ F, G\in \mbox{$\cal F$},\ F\neq G\}. $$ $\mbox{$\cal F$}$ is…

For an integer $k\ge2$, a tuple of $k$ positive integers $(M_i)_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \[ \sigma(M_1)=\cdots=\sigma(M_k)=M_1+\cdots+M_k \] holds. This is a generalization of amicable pairs. An amicable…

We improve our earlier upper bound on the numbers of antipodal pairs of points among $n$ points in ${\mathbb{R}}^3$, to $2n^2/5+O(n^c)$, for some $c<2$. We prove that the minimal number of antipodal pairs among $n$ points in convex position…

A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…

In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of…

The following is shown : Let $S=\{a_1,a_2,..,a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,*,\leq)$ with $a_1\leq a_2\leq...\leq a_{2n}$. Provided that a system of $n$ $a_{i_k} * a_{j_k}\ (a_{i_k}, a_{j_k} \in G ;\ 1…

Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and…

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^{\beta} n \big)$ as $n\to\infty$, where \begin{align*} \beta := \frac{1}{2} \log_2 \left(\frac{8 \pi…

A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado \cite{er} showed that a family of sets of size $n$ contains a…

In 1965, Bollob\'as proved that for a Bollob\'as set-pair system $\{(A_i,B_i)\mid i\in[m]\}$, the maximum value of $\sum_{i=1}^m\binom{|A_i|+|B_i|}{A_i}^{-1}$ is $1$. Heged\"{u}s and Frankl recently extended the concept of Bollob\'as…

In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…

Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show that the number of gauge groups of principal $G$-bundles over $B$ is finite up to $A_n$-equivalence for $n<\infty$. As an example, we give a lower bound of the…

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union.…