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: 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.

翻译：Matoušek 定理断言：对于任意 $k \ge 2$，任何碎裂函数为 $o(n^k)$ 的集合系统都满足分数 Helly 性质：在其 $k$ 元交超图中，正密度蕴含存在线性大小的团。Kalai 与 Meshulam 猜想该现象可推广至同调碎裂函数。该猜想已在同调碎裂函数有界、且基础集具有禁止同调子式（根据 van Kampen-Flores 定理的同调类比，这包括 $\mathbb{R}^d$）的集合系统中得到验证。我们在此研究方向提出两项贡献：- 我们研究了特定流形（可能带边界）上的同调子式，并证明了 van Kampen-Flores 定理与 Hanani-Tutte 定理的类比结果。- 我们引入了集合系统的 Radon 数与 Helly 数的分级类比，并将其增长率与原始参数相关联。这使得对于同调碎裂函数增长足够缓慢的情形，能够扩展 Kalai-Meshulam 猜想的验证范围。