Related papers: Characterizing 3-sets in Union-Closed Families

The Kneser--Poulsen Conjecture states that if the centers of a family of $N$ unit balls in ${\mathbb E}^d$ is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of…

Chv\'{a}tal conjectured that a star is amongst the largest intersecting subfamiles of a finite subset-closed family of sets. Kleitman later strengthened Chv\'{a}tal's conjecture, suggesting that maximal intersecting subfamilies of $2^{[n]}$…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

A family of sets $\mathcal{F} \subseteq 2^{[n]}$ is defined to be $l$-trace $k$-Sperner if for any $l$-subset $L$ of $[n]$ the family of traces $\mathcal{F}|_L=\{F \cap L: F \in \mathcal{F}\}$ does not contain any chain of length $k+1$. In…

A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of…

We consider the problem of minimizing the size of a family of sets G such that every subset of 1,...,n can be written as a disjoint union of at most k members of G, where k and n are given numbers. This problem originates in a real-world…

We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in…

We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture…

We say that a family $\mathcal F$ of $k$-element sets is a {\it $j$-junta} if there is a set $J$ of size $j$ such that, for any $F$, its presence in $\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary families by…

We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we…

Given a family $\mathcal F\subset 2^{[n]}$, its diversity is the number of sets not containing an element with the highest degree. The concept of diversity has proven to be very useful in the context of $k$-uniform intersecting families. In…

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi…

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most…

Let $M$ be a non-zero binary matrix with distinct rows where the rows are closed under certain logical operators. In this article, we investigate the existence of columns containing an equal or greater number of ones than zeros.…

A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the intersection of every pair of its elements has size at least $t$. A set system $\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length…

Let ${\mathcal G}$ be a family of subsets of an $n$-element set. The family ${\mathcal G}$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in ${\mathcal G}$ is of size at least $t$. For a real number $p\in(0,1)$…

A balanced pattern of order $2d$ is an element $P \in \{+,-\}^{2d}$, where both signs appear $d$ times. Two sets $A,B \subset [n]$ form $P$-pattern, which we denote by $\operatorname{pat}(A,B) = P$, if $A\triangle B = \{j_1,\ldots…

The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$…

In its usual form, Freiman's 3k-4 theorem states that if A and B are subsets of the integers of size k with small sumset (of size close to 2k) then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this…

A collection of $k$ sets is said to form a $k$-sunflower, or $\Delta$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ sunflower-free if it contains no sunflowers.…