Intersection patterns of set systems on manifolds with slowly growing homological shatter functions
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| Format: | Preprint |
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2026
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| author | Avvakumov, Sergey Bin, Marguerite Goaoc, Xavier |
| author_facet | Avvakumov, Sergey Bin, Marguerite Goaoc, Xavier |
| contents | A theorem of Matoušek asserts that for any $k \ge 2$, any set system whose shatter function is $o(n^k)$ enjoys a fractional Helly theorem of order $k$: in the $k$-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and ground set with a forbidden homological minor (which includes $\mathbb{R}^d$ by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research:
- We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem.
- We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture for sufficiently slowly growing homological shatter functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_02920 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intersection patterns of set systems on manifolds with slowly growing homological shatter functions Avvakumov, Sergey Bin, Marguerite Goaoc, Xavier Computational Geometry Discrete Mathematics F.2.2 A theorem of Matoušek asserts that for any $k \ge 2$, any set system whose shatter function is $o(n^k)$ enjoys a fractional Helly theorem of order $k$: in the $k$-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and ground set with a forbidden homological minor (which includes $\mathbb{R}^d$ by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture for sufficiently slowly growing homological shatter functions. |
| title | Intersection patterns of set systems on manifolds with slowly growing homological shatter functions |
| topic | Computational Geometry Discrete Mathematics F.2.2 |
| url | https://arxiv.org/abs/2601.02920 |