Related papers: An $(\aleph_0,k+2)$-Theorem for $k$-Transversals

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F $ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal…

Two celebrated extensions of Helly's theorem are the Fractional Helly theorem of Katchalski and Liu (1979) and the Quantitative Volume theorem of B\'ar\'any, Katchalski, and Pach (1982). Improving on several recent works, we prove an…

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an…

Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$…

A family of sets $F$ is said to satisfy the $(p,q)$ property if among any $p$ sets in $F$, some $q$ have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any $p \geq q \geq d+1$ there exists $c=c_d(p,q)$, such…

A family of sets satisfies the $(p,q)$ property if among every $p$ members of it some $q$ intersect. Given a number $0<r\le 1$, a set $S\subset \mathbb{R}^2$ is called $r$-fat if there exists a point $c\in S$ such that $B(c,r) \subseteq…

We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R^d is k-Leray where k depends on…

A collection of sets $\Fscr$ has the $(p,q)$-property if out of every $p$ elements of $\Fscr$ there are $q$ that have a point in common. A transversal of a collection of sets $\Fscr$ is a set $A$ that intersects every member of $\Fscr$.…

Given a poset $P$ we say a family $\mathcal{F}\subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, among all families of…

In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a…

A family of axis-aligned boxes in $\er^d$ is \emph{$k$-neighborly} if the intersection of every two of them has dimension at least $d-k$ and at most $d-1$. Let $n(k,d)$ denote the maximum size of such a family. It is known that $n(k,d)$ can…

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further…

Let $d\geq 2$ be an integer, $K_0$ a perfect field such that $char(K_0)$ does not divide $d$, $n > d$ an integer prime to $d$, $f(x)\in K_0[x]$ a degree $n$ monic polynomial without repeated roots, and $C_{f,d}$ a smooth projective model of…

We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a…

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of…

The aperture angle alpha(x, Q) of a point x not in Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the…

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 $F_1\in\mathscr{F}_1$…

Let $p$ denote the characteristic of ${\mathbb F}_q$, the finite field with $q$ elements. We prove that if $q$ is odd then an arc of size $q+2-t$ in the projective plane over ${\mathbb F}_q$, which is not contained in a conic, is contained…

Hadwiger and Debrunner showed that for families of convex sets in $\mathbb{R}^d$ with the property that among any $p$ of them some $q$ have a common point, the whole family can be stabbed with $p-q+1$ points if $p \geq q \geq d+1$ and…

In this note we study a conjecture by Jer\'onimo-Castro, Magazinov and Sober\'on which generalized a question posed by Dol'nikov. Let $F_1,F_2,\dots,F_n$ be families of translates of a convex compact set $K$ in the plane so that each two…