Related papers: Stability for t-intersecting families of permutati…

Let $t\ge 1$ be a given integer. Let ${\cal F}$ be a family of subsets of $[m]=\{1,2,\ldots,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$ with $|S|=|T|=k$, there do not exist $2t$ sets in ${\cal F}$ where $t$ subsets of…

A family $\mathcal{F} \subset \mathcal{P}(n)$ is $r$-wise $k$-intersecting if $|A_1 \cap \dots \cap A_r| \geq k$ for any $A_1, \dots, A_r \in \mathcal{F}$. It is easily seen that if $\mathcal{F}$ is $r$-wise $k$-intersecting for $r \geq 2$,…

We prove a stability version of a general result that bounds the permanent of a matrix in terms of its operator norm. More specifically, suppose $A$ is an $n \times n$ matrix over $\mathbb{C}$ (resp. $\mathbb{R}$), and let $\mathcal{P}$…

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote…

Determining the maximum size of a $t$-intersecting code in $[m]^n$ was a longstanding open problem of Frankl and F\"uredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting…

Two families of sets $\mathcal{A}$ and $\mathcal{B}$ are called cross-$t$-intersecting if $|A\cap B|\ge t$ for all $A\in \mathcal{A}$, $B\in \mathcal{B}$. An active problem in extremal set theory is to determine the maximum product of sizes…

It is well known that an intersecting family of subsets of an n-element set can contain at most 2^(n-1) sets. It is natural to wonder how `close' to intersecting a family of size greater than 2^(n-1) can be. Katona, Katona and Katona…

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its…

The matching number of a family of subsets of an $n$-element set is the maximum number of pairwise disjoint sets. The families with matching number $1$ are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the…

Let $\mathfrak{i}$ denote the minimal cardinality of a maximal independent family and let $\mathfrak{a}_T$ denote the minimal cardinality of a maximal family of pairwise almost disjoint subtrees of $2^{<\omega}$. Using a countable support…

Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erd\H{o}s-Ko-Rado theorem. Their intersection theorem says that the maximum size of a…

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a…

In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as…

Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is…

A family $\mathcal{A}$ of sets is {\it $t$-intersecting} if the cardinality of the intersection of every pair of sets in $\mathcal{A}$ is at least $t$, and is an {\it $r$-family} if every set in $\mathcal{A}$ has cardinality $r$. A…

Let $G$ be a permutation group on a set $\Omega$ of size $t$. We say that $\Lambda\subseteq\Omega$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We define the…

Costas arrays have been an interesting combinatorial object for decades because of their optimal aperiodic auto-correlation properties. Meanwhile, it is interesting to find families of Costas arrays or extended arrays with small maximal…

Given a point set $S$ in $\mathbb{R}^d$, a family of sets is $S$-intersecting if its members have a point in common in $S$. Recently, Edwards and Sober\'{o}n proved a fractional version of Halman's theorem for axis-parallel boxes, showing…

Twisting a given knot $K$ about an unknotted circle $c$ a full $n \in \mathbb{N}$ times, we obtain a "twist family" of knots $\{ K_n \}$. Work of Kouno-Motegi-Shibuya implies that for a non-trivial twist family the crossing numbers…

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is…