Optimal bounds for the colorful fractional Helly theorem

Abstract

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.