Related papers: Transversals to colorful intersecting convex sets

We prove an infinite $(p,q)$-theorem for piercing fat compact convex sets in $\RR^d$ with $k$-flats. Additionally, we develop a new framework through which infinite $(p,q)$-theorems concerning compact sets and $k$-flats can be extended to…

For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$,…

A \textit{$k$-transversal} to family of sets in $\mathbb{R}^d$ is a $k$-dimensional affine subspace that intersects each set of the family. In 1957 Hadwiger provided a necessary and sufficient condition for a family of pairwise disjoint,…

For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If…

Denote the collection of all $k$-flats in $AG(n,\mathbb{F}_q)$ by $\mathscr{M}(k,n)$. Let $\mathscr{F}_1\subset\mathscr{M}(k_1,n)$ and $\mathscr{F}_2\subset\mathscr{M}(k_2,n)$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$…

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…

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of…

Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a…

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…

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erd\H{o}s and Gr\"unbaum. Namely, if in an…

Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$,…

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…

Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$, $\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections $F\cap G$ with $F\in…

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 $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in…

B\'ar\'any, Katchalski and Pach proved the following quantitative form of Helly's theorem. If the intersection of a family of convex sets in $\mathbb{R}^d$ is of volume one, then the intersection of some subfamily of at most $2d$ members is…

We prove that every pointed closed convex set in $\mathbb{R}^n$ is the intersection of all the rational closed halfspaces that contain it. This generalizes a previous result by the authors for compact convex sets.

We say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects any other set in…

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that…

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,…