A fractional Helly theorem for set systems with slowly growing homological shatter function

Marguerite Bin

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arXiv · 2411.18605

A fractional Helly theorem for set systems with slowly growing homological shatter function

Marguerite Bin

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Abstract

We study parameters of the convexity spaces associated with families of sets in $\mathbb{R}^d$ where every intersection between $t$ sets of the family has its Betti numbers bounded from above by a function of $t$. Although the Radon number of such families may not be bounded, we show that these families satisfy a fractional Helly theorem. To achieve this, we introduce graded analogues of the Radon and Helly numbers. This generalizes previously known fractional Helly theorems.

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Marguerite Bin. 2024-11-27. A fractional Helly theorem for set systems with slowly growing homological shatter function. https://arxiv.org/abs/2411.18605

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