Related papers: The Hajnal--Rothschild problem

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $|\cap \{F\colon F\in \mathcal{F}\}|<t$, $|\cap…

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 say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be \emph{cross-$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects…

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. Families $\mathcal{A}_1, \mathcal{A}_2, \dots, \mathcal{A}_k$ of sets are cross-$t$-intersecting if, for every $i$ and $j$ in $\{1, 2, \dots, k\}$ with $i…

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\'a}lv{\"o}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in…

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ elements of $[m]$ with repetition and without ordering. We use $\left(\binom {[m]}{k}\right)$ to denote all the $k$-multisets of $[m]$. Two multiset…

Let $\mathcal{F}$ be a family of $k$-sized subsets of $[n]$ that does not contain $s$ pairwise disjoint subsets. The Erd\H{o}s Matching Conjecture, a celebrated and long-standing open problem in extremal combinatorics, asserts the maximum…

In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\leq i\leq r$,…

Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$…

We consider families of $k$-subsets of $\{1, \dots, n\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\mathcal{F}$ to have no perfect matching is for there to be a blocking set,…

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$.…

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\mathcal{A}, \mathcal{B}…

For a family $\mathcal F$, let $\mathcal D(\mathcal F)$ stand for the family of all sets that can be expressed as $F\setminus G$, where $F,G\in \mathcal F$. A family $\mathcal F$ is intersecting if any two sets from the family have…

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…

Let $\mathcal{F}$ be a family of subsets of $[n]$. The diameter of $\mathcal{F}$ is the maximum size of symmetric differences among pairs of its members. Resolving a conjecture of Erd\H{o}s, Kleitman determined the maximum size of a family…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-$t$-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$ in at least $t$ elements. An active problem in extremal set theory is to determine…

Ever since the famous Erd\H{o}s-Ko-Rado theorem initiated the study of intersecting families of subsets, extremal problems regarding intersecting properties of families of various combinatorial objects have been extensively investigated.…

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…

For integers $n\ge s\ge2$, let $e(n,s)$ denote the maximum size of a family $\F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is the non-uniform…

Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as…