Related papers: Forbidding intersection patterns between layers of…

Let $S_{n}$ denote the set of permutations of $[n]=\{1,2,\dots, n\}$. For each integer $k\geq 1$, let $S_{n,k}$ be the set of all permutations of $[n]$ with exactly $k$ disjoint cycles. A subset $H\subseteq S_{n,k}$ is to be a matching if…

A family of sets is called $r$-\emph{cover free} if no set in the family is contained in the union of $r$ (or less) other sets in the family. A $1$-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's…

Let \cal A be a family of subsets of an n-set such that \cal A does not contain distinct sets A and B with |A\B| = 2|B\A|. How large can \cal A be? Our aim in this note is to determine the maximum size of such an \cal A. This answers a…

A family $\F$ of sets is said to be intersecting if any two sets in $\F$ have nonempty intersection. The celebrated Erd{\H o}s-Ko-Rado theorem determines the size and structure of the largest intersecting family of $k$-sets on an $n$-set…

A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$ for any $A,B \in F$. The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size $f(n,k,t)$ of a $t$-intersecting family of…

The maximum size of $t$-intersecting families is one of the most celebrated topics in combinatorics, and its size is known as the Erd\H{o}s-Ko-Rado theorem. Such intersecting families, also known as constant-weight anticodes in coding…

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union.…

A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado \cite{er} showed that a family of sets of size $n$ contains a…

We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all…

For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family…

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…

For a given positive integer $k$ we say that a family of subsets of $[n]$ is $k$-antichain saturated if it does not contain $k$ pairwise incomparable sets, but whenever we add to it a new set, we do find $k$ such sets. The size of the…

For a finite poset $P$, we study the expected size of the intersection of two independent uniformly random antichains. Equivalently, we evaluate the sum of $|A\cap A'|$ over all ordered pairs of antichains. For general posets this statistic…

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

A pair of families $(\cF,\cG)$ is said to be \emph{cross-Sperner} if there exists no pair of sets $F \in \cF, G \in \cG$ with $F \subseteq G$ or $G \subseteq F$. There are two ways to measure the size of the pair $(\cF,\cG)$: with the sum…

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…

A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets…

An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue)…

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran…