Optimal Bounds for the Colorful Fractional Helly Theorem

The well known fractional Helly theorem and colorful Helly theorem can be merged into the so called colorful fractional Helly theorem. It states: For every $\alpha \in (0, 1]$ and every non-negative integer $d$, there is $\beta_{col} = \beta_{col}(\alpha, d) \in (0, 1]$ with the following property. Let $\mathcal{F}_1, \dots, \mathcal{F}_{d+1}$ be finite nonempty families of convex sets in $\mathbb{R}^d$ of sizes $n_1, \dots, n_{d+1}$ respectively. If at least $\alpha n_1 n_2 \cdots n_{d+1}$ of the colorful $(d+1)$-tuples have a nonempty intersection, then there is $i \in [d+1]$ such that $\mathcal{F}_i$ contains a subfamily of size at least $\beta_{col} n_i$ with a nonempty intersection. (A colorful $(d+1)$-tuple is a $(d+1)$-tuple $(F_1, \dots , F_{d+1})$ such that $F_i$ belongs to $\mathcal{F}_i$ for every $i$.) The colorful fractional Helly theorem was first stated and proved by B\'ar\'any, Fodor, Montejano, Oliveros, and P\'or in 2014 with $\beta_{col} = \alpha/(d+1)$. In 2017 Kim proved the theorem with better function $\beta_{col}$, which in particular tends to $1$ when $\alpha$ tends to $1$. Kim also conjectured what is the optimal bound for $\beta_{col}(\alpha, d)$ and provided the upper bound example for the optimal bound. The conjectured bound coincides with the optimal bounds for the (non-colorful) fractional Helly theorem proved independently by Eckhoff and Kalai around 1984. We verify Kim's conjecture by extending Kalai's approach to the colorful scenario. Moreover, we obtain optimal bounds also in more general setting when we allow several sets of the same color.Comment: 13 pages, 1 figure. The main technical result is extended to c colors, where c is a positive integer, in contrast to the previous version where we only allowed (d+1) colors. We added the acknowledgment

Theorems of Carathéodory, Helly, and Tverberg without dimension

We initiate the study of no-dimensional versions of classical theorems in convexity. One example is Carathéodory’s theorem without dimension: given an n-element set P in a Euclidean space, a point a∈convP, and an integer r≤n, there is a subset Q⊂P of r elements such that the distance between a and convQ is less than diamP/2r−−√. In an analoguos no-dimension Helly theorem a finite family F of convex bodies is given, all of them are contained in the Euclidean unit ball of Rd. If k≤d, |F|≥k, and every k-element subfamily of F is intersecting, then there is a point q∈Rd which is closer than 1/k−−√ to every set in F. This result has several colourful and fractional consequences. Similar versions of Tverberg’s theorem and some of their extensions are also established

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry