Related papers: Incompatible intersection properties

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.…

It is demonstrated that each nearly neighbourly family of standard boxes in $\mathbb{R}^3$ has at most 12 elements. A combinatorial classification of all such families that have exactly 12 elements is given. All families satisfying an extra…

We show that the biggest possible average set size in the complement $2^{\{1,2,\ldots, n\}} \setminus A$ of a union-closed family $A \subset 2^{\{1,2, \ldots, n\}}$ is $\tfrac{n+1}{2}$. With the same proof we get a sharp upper bound for the…

Let M be a compact, orientable, irreducible, atoroidal 3-manifold with boundary an incompressible torus. Techniques based on the characteristic submanifold theory are used to bound the intersection number of two slopes \alpha and \beta on…

A family $\mathcal F$ has covering number $\tau$ if the size of the smallest set intersecting all sets from $\mathcal F$ is equal to $\tau$. Let $M(n,k,\tau)$ stand for the size of the largest intersecting family $\mathcal F$ of $k$-element…

A family $\mbox{$\cal F$}=\{F_1,\ldots,F_m\}$ of subsets of $[n]$ is said to be ordered, if there exists an $1\leq r\leq m$ index such that $n\in F_i$ for each $1\leq i\leq r$, $n\notin F_i$ for each $i>r$ and $|F_i|\leq |F_j|$ for each…

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…

For any positive integers $k,r,n$ with $r \leq \min\{k,n\}$, let $\mathcal{P}_{k,r,n}$ be the family of all sets $\{(x_1,y_1), \dots, (x_r,y_r)\}$ such that $x_1, \dots, x_r$ are distinct elements of $[k] = \{1, \dots, k\}$ and $y_1, \dots,…

Two subsets $A,B$ of an $n$-element ground set $X$ are said to be \emph{crossing}, if none of the four sets $A\cap B$, $A\setminus B$, $B\setminus A$ and $X\setminus(A\cup B)$ are empty. It was conjectured by Karzanov and Lomonosov forty…

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each…

Matroid is a generalization of many fundamental objects in combinatorial mathematics , and matroid intersection problem is a classical subject in combinatorial optimization . However , only the intersection of two matroids are well…

A family $\mathcal C$ of sets is hereditary if whenever $A\in \mathcal C$ and $B\subset A$, we have $B\in \mathcal C$. Chv\'atal conjectured that the largest intersecting subfamily of a hereditary family is the family of all sets containing…

The celebrated Erd\H{o}s-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona,…

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…

Given a set of points in the plane, a \emph{crossing family} is a collection of segments, each joining two of the points, such that every two segments intersect internally. Aronov et al. [Combinatorica,~14(2):127-134,~1994] proved that any…

Let $\mathcal{S}$ be a family of sets with VC-codensity less than $2$. We prove that, if $\mathcal{S}$ has the $(\omega, 2)$-property (for any infinitely many sets in $\mathcal{S}$, at least $2$ among them intersect), then $\mathcal{S}$ can…

We consider the problem of determining the maximum number of pairs $F\subseteq F'$ in a family $\mathcal{F}\subseteq 2^{[n]}$ that avoids certain posets $P$ of height 2. We show that for any such $P$ the number of pairs is…

Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present…

The Union-Closed Sets Conjecture asks whether every union-closed set family $\mathcal{F}$ has an element contained in half of its sets. In 2022, Nagel posed a generalisation of this problem, suggesting that the $k$th-most popular element in…

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical…