On Helly number for crystals and cut-and-project sets

We prove existence of finite Helly numbers for crystals and for cut-and-project sets with convex windows; also we prove exact bound of $k+6$ for the Helly number of a crystal consisting of $k$ copies of a single lattice. We show that there are sets of finite local complexity that do not have finite Helly numbers

Helly-type problems

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals

Bounding Radon's number via Betti numbers

We prove general topological Radon type theorems for sets in $\mathbb R^d$, smooth real manifolds or finite dimensional simplicial complexes. Combined with a recent result of Holmsen and Lee, it gives fractional Helly and colorful Helly theorems, and consequently an existence of weak $\varepsilon$-nets as well as a $(p,q)$-theorem. More precisely: Let $X$ be either $\mathbb R^d$, smooth real $d$-manifold, or a finite $d$-dimensional simplicial complex. Then if $\mathcal F$ is a finite family of sets in $X$ such that $\widetilde\beta_i(\bigcap \mathcal G; \mathbb Z_2)$ is at most $b$ for all $i=0,1,\ldots, k$ and $\mathcal G\subseteq \mathcal F$, then the Radon's number of $\mathcal F$ is bounded in terms of $b$ and $X$. Here $k=\left\lceil\frac{d}{2}\right\rceil-1$ if $X=\mathbb R^d$; $k=d-1$ if $X$ is a smooth real $d$-manifold and not a surface, $k=0$ if $X$ is a surface and $k=d$ if $X$ is a $d$-dimensional simplicial complex. Using the recent result of the author and Kalai, we manage to prove the following optimal bound on fractional Helly number for families of open sets in a surface: Let $\mathcal F$ be a finite family of open sets in a surface $S$ such that for every $\mathcal G\subseteq \mathcal F$, $\bigcap \mathcal G$ is either empty, or path-connected. Then the fractional Helly number of $\mathcal F$ is at most three. This also settles a conjecture of Holmsen, Kim, and Lee about an existence of a $(p,q)$-theorem for open subsets of a surface.Comment: 11 pages, 2 figure

Quantitative Fractional Helly and (p,q)(p,q)-Theorems

We consider quantitative versions of Helly-type questions, that is, instead of finding a point in the intersection, we bound the volume of the intersection. Our first main geometric result is a quantitative version of the Fractional Helly Theorem of Katchalski and Liu, the second one is a quantitative version of the $(p,q)$-Theorem of Alon and Kleitman.Comment: 11 page