Related papers: Classification of maximum hittings by large famili…

A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$…

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following…

A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in…

We study the problem of determining the size of the largest intersecting $P$-free family for a given partially ordered set (poset) $P$. In particular, we find the exact size of the largest intersecting $B$-free family where $B$ is the…

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 $r \geq 2$,…

A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichain if $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor…

A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while…

Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ with distance set $D$. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B…

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…

A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest…

This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H.…

We show that an $n$-uniform maximal intersecting family has size at most $e^{-n^{0.5+o(1)}}n^n$. This improves a recent bound by Frankl. The Spread Lemma of Alweiss, Lowett, Wu and Zhang plays an important role in the proof.

Consider the random set system of {1,2,...,n}, where each subset in the power set is chosen independently with probability p. A set H is said to be a hitting set if it intersects each chosen set. The second moment method is used to exhibit…

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1,…

A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erd\H{o}s, Ko, and Rado states that every intersecting family of r-sets…

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. We say that $(\mathcal{F}_1, \dots,…

Let $V$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$. In this paper we describe the structure of maximal non-trivial $t$-intersecting families of $k$-dimensional subspaces of $V$ with large size. We also determine…

A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each…

Two families of sets \(\mathcal{A}\) and \(\mathcal{B}\) are called \emph{cross-\(t\)-intersecting} if \(|A \cap B| \geq t\) for all \(A \in \mathcal{A}\) and \(B \in \mathcal{B}\). Determining the maximum product of sizes for such…