Related papers: Probably Intersecting Families are Not Nested

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…

Resolving a conjecture of Methuku and the first author we determine the size of the largest family of subsets of an $n$-element set avoiding both $Y_k$ and $Y_k'$ as induced subposets. The result follows as a consequence of the analogous…

Let $n$, $r$, and $k$ be positive integers such that $k, r \geq 2$, $L$ a non-empty subset of $[k]$, and $\mathcal{F}_i \subseteq \binom{[n]}{k}$ for $1 \leq i \leq r$. We say that non-empty families $\mathcal{F}_1, \mathcal{F}_2, \ldots,…

Intersecting families and blocking sets feature prominently in extremal combinatorics. We examine the following generalization of an intersecting family investigated by Hajnal, Rothschild, and others. If $s \geq 1$, $k \geq 2$, and $u \geq…

Let $V$ be a finite dimensional vector space over a finite field, and $\mathcal{F}$ a family consisting of $k$-subspaces of $V$. The family $\mathcal{F}$ is called $t$-intersecting if $\dim(F_{1}\cap F_{2})\geq t$ for any $F_{1}, F_{2}\in…

A family $\mathcal F\subset {[n]\choose k}$ is $U(s,q)$ of for any $F_1,\ldots, F_s\in \mathcal F$ we have $|F_1\cup\ldots\cup F_s|\le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number…

Let $k$ and $m$ be positive integers. A collection of $k$-multisets from $\{1,..., m \}$ is intersecting if every pair of multisets from the collection is intersecting. We prove that for $m \geq k+1$, the size of the largest such collection…

A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does not contain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and…

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…

We make some progress on a question of Babai from the 1970s, namely: for $n, k \in \mathbb{N}$ with $k \le n/2$, what is the largest possible cardinality $s(n,k)$ of an intersecting family of $k$-element subsets of $\{1,2,\ldots,n\}$…

We show that, for $pn \to \infty$, the largest set in a $p$-random sub-family of the power set of $\{1, \ldots, n\}$ containing no $k$-chain has size $( k - 1 + o(1) ) p \binom{n}{n/2}$ with high probability. This confirms a conjecture of…

The celebrated Erd\H{o}s-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform…

This paper resolves two open problems in extremal set theory. For a family $\mathcal{F} \subseteq 2^{[n]}$ and $i, j\in [n]$, we denote $\mathcal{F} (i,\bar{j})=\{F\backslash\{i\}: F\in \mathcal{F}, F\cap\{i,j\}=\{i\}\}$. The sturdiness…

Let $n=2k+1$ be odd with $k\geq3$. In this note, we give two intersecting families with diversity larger than $\sum_{i=k+1}^{2k} \binom {2k}{i}$, which disprove a conjecture of Huang.

Let $\mathcal A=\{A_1,\ldots,A_n\}$ be a family of sets in the plane. For $0 \leq i < n$, denote by $f_i$ the number of subsets $\sigma$ of $\{1,\ldots,n\}$ of cardinality $i+1$ that satisfy $\bigcap_{i \in \sigma} A_i \neq \emptyset$. Let…

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the…

A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A…

A family of subsets of $[n]$ is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in…

In this note, we present a conjecture on intersections of set families, and a rephrasing of the conjecture in terms of principal downsets of Boolean lattices. The conjecture informally states that, whenever we can express the measure of a…

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…