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• W3136361164 abstract "Intersection patterns of convex sets in $mathbb{R}^d$ have the remarkable property that for $d+1 le k le ell$, in any sufficiently large family of convex sets in $mathbb{R}^d$, if a constant fraction of the $k$-element subfamilies have nonempty intersection, then a constant fraction of the $ell$-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system $mathcal{F}$ in $mathbb{R}^d$. Quantitatively, our bounds depend on how complicated the intersection of $ell$ elements of $mathcal{F}$ can be, as measured by the sum of the $lceilfrac{d}2rceil$ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to $d+1$. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matousek and Nivash to recast a simplicial complex as a homological minor of a cubical complex." @default.
• W3136361164 created "2021-03-29" @default.
• W3136361164 creator A5021289985 @default.
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• W3136361164 date "2021-03-16" @default.
• W3136361164 modified "2023-09-27" @default.
• W3136361164 title "A Stepping-Up Lemma for Topological Set Systems" @default.
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