Related papers: On $L$-close Sperner systems

The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set…

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S={F \cap S : F \in \mathcal{F}}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$…

We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant…

The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022,…

A central result in extremal set theory is the celebrated theorem of Sperner from 1928, which gives the size of the largest family of subsets of [n] not containing a 2-chain. Erdos extended this theorem to determine the largest family…

A family $\mathcal{F}\subset 2^G$ of subsets of an abelian group $G$ is a Sidon system if the sumsets $A+B$ with $A,B\in \mathcal{F}$ are pairwise distinct. Cilleruelo, Serra and the author previously proved that the maximum size $F_k(n)$…

Let $A \subseteq F_2^n$ be a set with $|2A| = K|A|$. We prove that if (1) for at least a fraction $1-K^{-9}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at most $L\cdot|A|/K$, or (2) for at least a fraction $K^{-L}$ of all $s \in…

A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and…

A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set…

A family of sets is said to be \emph{intersecting} if any two sets in the family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of determining the maximum possible size of a union of $r$ different intersecting families of…

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…

Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner…

Sperner's bound on the size of an antichain in the lattice P(S) of subsets of a finite set S has been generalized in three different directions: by Erdos to subsets of P(S) in which chains contain at most r elements; by Meshalkin to certain…

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power…

The two part Sperner theorem of Katona and Kleitman states that if $X$ is an $n$-element set with partition $X_1 \cup X_2$, and $\cF$ is a family of subsets of $X$ such that no two sets $A, B \in \cF$ satisfy $A \subset B$ (or $B \subset…

A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erd\H{o}s characterised the largest families of subsets of…

We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

For a positive integer $d$, a set of points in $d$-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let $f(d)$ denote the largest size of an almost-equidistant set…

Let $\mathcal{F}$ be a family of subsets of $[n]=\{1,\ldots,n\}$ and let $L$ be a set of nonnegative integers. The family $\mathcal{F}$ is \emph{$L$-intersecting} if $|F\cap F'|\in L$ for every two distinct members $F,F'\in\mathcal{F}$; and…