Related papers: Variations on the Bollob\'as set-pair theorem

Optimal gap sequences in Shellsort, defined as gap sequences having the minimised maximum number of comparisons for a fixed number of pairwise distinct elements, are found by minimax search in reduced permutational spaces, namely Bad…

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.…

We study pairs of subsets $A, B$ of a compact abelian group $G$ where the sumset $A+B:=\{a+b: a\in A, b\in B\}$ is small. Let $m$ and $m_{*}$ be Haar measure and inner Haar measure on $G$, respectively. Given $\varepsilon>0$, we classify…

Proposals for Bell inequality tests on systems restricted by superselection rules often require operations that are difficult to implement in practice. In this paper, we derive a new Bell inequality, where pairs of states are used to…

An overlap representation is an assignment of sets to the vertices of a graph in such a way that two vertices are adjacent if and only if the sets assigned to them overlap. The overlap number of a graph is the minimum number of elements…

Let $P:\mathbb{C}^n\rightarrow \mathbb{C}$ be an $m$-homogeneous polynomial given by \[P(x)= \sum_{1\leq j_1\leq \ldots \leq j_m \leq n} c_{j_1 \ldots j_m} x_{j_1}\ldots x_{j_m}.\] Defant and Schl\"uters defined a non-symmetric associated…

Let $A \subset \mathbb{R}$ be finite. We quantitatively improve the Balog-Wooley decomposition, that is $A$ can be partitioned into sets $B$ and $C$ such that $$\max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) ,…

A set $U$ of unit vectors is selectively balancing if one can find two disjoint subsets $U^+$ and $U^-$, not both empty, such that the Euclidean distance between the sum of $U^+$ and the sum of $U^-$ is smaller than $1$. We prove that the…

In this paper, we prove the twin prime conjecture showing that \begin{align} \sum \limits_{\substack{p\leq x\\p,p+2\in \mathbb{P}}}1\geq (1+o(1))\frac{x}{2\mathcal{C}\log^2 x}\nonumber \end{align} where $\mathcal{C}:=\mathcal{C}(2)>0$ fixed…

We provide a version of the celebrated theorem of Koml\'os in which, rather then random quantities, a sequence of finitely additive measures is considered. We obtain a form of the subsequence principle and some applications.

We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the total-variation distance between…

Let $G$ be an additive abelian group. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there are $\varepsilon_1, \ldots, \varepsilon_{\ell} \in \{-1, 1\}$ such that…

Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating…

A sequence of $d+1$ signs $+$ and $-$ beginning with a $+$ is called a {\em sign pattern (SP)}. We say that the real polynomial $P:=x^d+\sum _{j=0}^{d-1}a_jx^j$, $a_j\neq 0$, defines the SP $\sigma :=(+$,sgn$(a_{d-1})$, $\ldots$,…

An additive system for the nonnegative integers is a family (A_i)_{i\in I} of sets of nonnegative integers with 0 \in A_i for all i \in I such that every nonnegative integer can be written uniquely in the form \sum_{i\in I} a_i with a_i \in…

The Bell theorem for a pair of two-state systems in a singlet state is formulated for the entire range of measurement settings.

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\ldots,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\geq sk+1$ for all $1\leq i \leq p$, this…

A tie-point of a compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. Set-theoretically a tie-point of N* is an ultrafilter whose dual maximal ideal can be…

Sellami and Sirvent conjectured that the balanced pair algorithm fails for the following pair of Pisot substitutions: \[ \varphi_0: \begin{array}{l} a \mapsto abc b \mapsto a c \mapsto ac \end{array} \quad \text{ and } \quad \varphi_1:…

Combining ideas of Pham, Sah, Sawhney, and Simkin on spread perfect matchings in super-regular bipartite graphs with an algorithmic blow-up lemma, we prove a spread version of the blow-up lemma. Intuitively, this means that there exists a…