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.

翻译：Matoušek定理断言：对任意$k \ge 2$，若某集系统的散射函数为$o(n^k)$，则该系统满足$k$阶分数Helly性质：在$k$元交超图中，正密度蕴含线性尺寸团。Kalai与Meshulam猜想这一现象可推广至同调散射函数。该猜想已在具有有界同调散射函数且底集禁止某同调子式（根据van Kampen-Flores定理的同调类比，这包括$\mathbb{R}^d$）的集系统中得到验证。我们为该研究方向贡献两项工作：- 研究特定流形（可能带边界）中的同调子式，并证明其上的van Kampen-Flores定理与Hanani-Tutte定理类比形式；- 引入集系统Radon数与Helly数的分次类比，并将其增长速率与原参数关联。这些结果使Kalai-Meshulam猜想的验证得以推广至足够慢增长的同调散射函数情形。