Related papers: Improved bounds on the Hadwiger-Debrunner numbers

The Hadwiger-Debrunner number $HD_d(p,q)$ is the minimal size of a piercing set that can always be guaranteed for a family of compact convex sets in $\mathbb{R}^d$ that satisfies the $(p,q)$ property. Hadwiger and Debrunner showed that…

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 $F$ of sets is said to satisfy the $(p,q)$-property if among any $p$ sets of $F$ some $q$ intersect. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that any family of compact convex sets in $\mathbb{R}^d$ that…

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…

A family $\mathcal{F}$ of sets satisfies the $(p,q)$-property if among every $p$ members of $\mathcal{F}$, some $q$ can be pierced by a single point. The celebrated $(p,q)$-theorem of Alon and Kleitman asserts that for any $p \geq q \geq…

A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. It is shown that for every $p\ge q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family $\scr F$ of…

Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G on vertex sets X,Y that satisfies the following condition; one can add the edges between X and Y that do not belong to G one after the other so that whenever a…

In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in $\mathbb R^d$. Very recently, this result was extended to the $(p,q)$ setting with $p \geq q \geq d+1$ by Edwards and Sober\'on, and subsequently to the case $p…

A finite family $\mathcal{F}$ of $d$-dimensional convex polytopes is called $k$-neighborly if $d-k\le\textup{dim}(C\cap C')\le d-1$ for any two distinct members $C,C'\in\mathcal{F}$. In 1997, Alon initiated the study of the general function…

Hadwiger's theorem is a Helly-type theorem involving common transversals to families of convex sets instead of common intersections. Subsequently, Pollack and Wenger identified a necessary and sufficient condition, called a consistent…

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,…

Let $P$ be a set of $n$ points in $\mathbb{R}^d$, in general position. We remove all of them one by one, in each step erasing one vertex of the convex hull of the current remaining set. Let $g_d(P)$ denote the number of different removal…

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…

A $d$-{\em interval} is a union of at most $d$ disjoint closed intervals on a fixed line. Tardos [Combinatorica 15 (1995), 123-134] and the second author [Disc. Comput. Geom. 18 (1997), 195-203] used topological tools to bound the…

We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive…

Hadwiger's transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane…

Hadwiger's transversal theorem gives necessary and sufficient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful…

We continue with the investigation began in "The Dixmier conjecture and the shape of possible counterexamples". In that paper we introduced the notion of an irreducible pair (P,Q) as the image of the pair (X,Y) of the canonical generators…

Motivated by questions about $\mathbb{C}_p$-valued Fourier transform on the locally compact group $(\mathbb{Q}_p^d,+)$, we study invariant norms on the $p$-adic Schr\"odinger representation of the Heisenberg group. Our main result is a…

Let $A\subseteq \mathbb{Z}_p^2$ be a set of size $2p+1$ for prime $p\geq 5$. In this paper, we prove that $A\hat{+}A=\{a_1+a_2\mid a_1,a_2\in A, a_1\neq a_2\}$ has cardinality at least $4p$. This result is the first advancement in over two…