Homological Shatter Functions

• Homological shatter functions are a topologically-enriched analog of classical shatter functions, quantifying complexity via Betti numbers in intersecting set systems.
• They extend key combinatorial theorems such as the fractional Helly and Radon theorems from finite-dimensional spaces to more general topological settings, including manifolds.
• Controlling the growth of these functions leads to novel graded parameters that bridge discrete geometry and algebraic topology, yielding strong intersection guarantees.

@@@@1@@@@ homological shatter function is a topologically-enriched analog of the classical shatter function from VC theory, used to quantify the complexity of intersection patterns in set systems within topological spaces. By controlling the growth of these functions—measured via Betti numbers of intersections—fundamental combinatorial phenomena such as the (fractional) Helly theorem, Radon’s lemma, and their graded analogues can be generalized from finite-dimensional geometric set systems to more general topological settings, including families of sets in manifolds. This extension plays a crucial role in identifying structural properties of set systems through homological constraints and leads to new connections between combinatorics, topology, and convexity theory (Avvakumov et al., 6 Jan 2026, Bin, 2024).

1. Definition and Notation

Given a set system $F$ in a topological space (e.g., $\mathbb{R}^d$ or a manifold) and an integer $h \geq 0$, the h-th homological shatter function is defined as

$$\phi_{F}^{(h)}(k) = \sup \left\{ \max_{0 \leq i \leq h} \tilde\beta_i \left( \bigcap_{A \in G} A ; \mathbb{Z}_2 \right) \;\Bigm|\; G \subseteq F, |G| \leq k \right\},$$

where $\tilde\beta_i(X; \mathbb{Z}_2)$ is the i-th reduced Betti number of $X$ with $\mathbb{Z}_2$-coefficients. This function records the maximal topological complexity (up to homology degree $h$) among all intersections of up to $k$ members of $F$ (Avvakumov et al., 6 Jan 2026, Bin, 2024).

Special cases:

• For $h = 0$, $\phi_{F}^{(0)}(k)$ counts connected components (as in classical VC dimension).
• For $h = \infty$, $\phi_{F}^{(\infty)} \equiv 0$ corresponds to acyclic covers.

The homological shatter function extends the classical shatter function $\pi_F(k)$, which is purely combinatorial.

2. Homological Shatter Functions and Fractional Helly Theorems

The Kalai–Meshulam conjecture posits that, for set systems in $\mathbb{R}^d$ with homological shatter function $\phi_F^{(d)}(n) = O(n^d)$, a $(d+1)$-fractional Helly theorem holds: a positive fraction of $(d+1)$-wise intersections implies the existence of a large subfamily with non-empty total intersection (Avvakumov et al., 6 Jan 2026).

Recent results verify this under subpolynomial or controlled (sufficiently slow) growth of the homological shatter function for systems whose ground space excludes certain forbidden homological minors. Specifically, for a fixed simplicial complex $K$ of dimension $d$, there exists a diverging function $\Psi_K$ such that if

$$\phi_F^{(d)}(t) \leq \Psi_K(t) \quad \text{(for all large } t\text{)},$$

and the ground space forbids $K$ as a homological minor, then $F$ has fractional Helly number at most $\mu(K) + 1$, where $\mu(K)$ is the sum of the dimensions of two disjoint simplices in $K$ (Avvakumov et al., 6 Jan 2026).

This result is robust under extremely slow (e.g., iterated logarithmic) divergence of $\phi_F^{(d)}$, and generalizes the classical result for polynomially-bounded combinatorial shatter functions.

3. Graded Radon and Helly Numbers

To analyze how intersection-combinatorial properties behave on small subfamilies, graded versions of Radon and Helly numbers are introduced. For $t \in \mathbb{N}$, define: $$r_F(t) := \sup\{ \mathrm{rad}_{F'} \mid F' \subseteq F,\, |F'| \leq t \}, \qquad h_F(t) := \sup\{ \mathrm{h}_{F'} \mid F' \subseteq F,\, |F'| \leq t \},$$ where $\mathrm{rad}_{F'}$ and $\mathrm{h}_{F'}$ are the usual Radon and Helly numbers of $F'$ (Avvakumov et al., 6 Jan 2026, Bin, 2024).

Key properties:

• Graded analogue of Levi’s inequality: $h_F(t) \leq r_F(t) - 1$ for all $t$.
• If $r_F(t) = o(\log_2 t)$ as $t \to \infty$, then the global Radon number is finite.
• If $h_F(t) < t$ for all $t > t_0$, then the (ungraded) Helly number is at most $t_0$.

Bounding the growth of $\phi_{F}^{(h)}$ controls the graded Radon and Helly numbers, thus translating topological complexity into combinatorial intersection theorems.

4. Verification of the Kalai–Meshulam Conjecture under Slow Growth

By combining the forbidden-minor technology (homological analogues of van Kampen–Flores and Hanani–Tutte theorems), the graded Radon/Helly framework, and combinatorial reduction techniques, it is established that any set system $F$ on a manifold, whose homological shatter function $\phi_F^{(d)}(t)$ grows more slowly than a diverging function $\Psi_K$ (as above), has a bounded fractional Helly number (Avvakumov et al., 6 Jan 2026).

In the case where the forbidden minor is $\Delta_{d+2}^{(\lceil d/2\rceil)}$, this yields a fractional Helly index of $d+1$ for set systems in $\mathbb{R}^d$ with slowly-growing $d$-th homological shatter function.

This verification supplies a topological strengthening of classical combinatorial theorems: as soon as the homological shatter function is controlled (even mildly), strong intersection properties must hold.

5. Topological Techniques and Proof Mechanisms

The proof framework deploys new topological results:

• A manifold version of the van Kampen–Flores theorem: for fixed $d$ and homological constraint $b$, there exists $N$ so that no nontrivial chain–map

$$C_*(\Delta_N^{(\lceil d/2\rceil)}) \rightarrow C_*(M)$$

(for any compact PL $d$-manifold $M$) can be an almost-embedding if $\beta_{\lceil d/2\rceil}(M) \leq b$.

• A homological Hanani–Tutte theorem for compact even-dimensional manifolds $M$, involving chain-maps $f$ where images of disjoint $k$-faces intersect in even cardinality.

These ingredients, combined with "color–Ramsey" arguments and careful analysis of skeletons in the nerve of $F$, translate topological non-embeddability into combinatorial intersection guarantees. This approach is central to bounding graded Radon numbers and deducing fractional Helly-type results (Avvakumov et al., 6 Jan 2026).

6. Illustrative Examples and Corollaries

The flexibility of the homological shatter framework is illustrated by the following:

• Extension from $\mathbb{R}^d$ to manifolds: Helly, Radon, $(p,q)$, and fractional Helly results carry over to manifolds with appropriate Betti constraints.
• Stably parallelizable and $\pi$-manifolds of dimension $2k$: such manifolds admit forbidden minors, enabling the machinery; examples include tori and orientable 3-manifolds.
• For semi-algebraic families in $\mathbb{R}^d$ of bounded complexity, $\phi_{F}^{(h)}$ is polynomially bounded, covering key instances in discrete geometry.
• Construction of set systems where $\phi_{F}^{(h)}(t)$ grows as slowly as desired (e.g., $O(\log\log t)$), establishes that the fractional Helly property persists under arbitrarily slowly growing homological shatter functions.
• Existence of families where graded Radon numbers grow only logarithmically, though the (ungraded) Radon number is infinite; thus, the slow growth of graded parameters suffices for fractional Helly-type theorems (Avvakumov et al., 6 Jan 2026, Bin, 2024).

7. Broader Significance and Connections

Homological shatter functions provide a robust framework bridging combinatorial geometry, algebraic topology, and the theory of convexity spaces. They generalize the classical VC-dimension paradigm, enabling the extension of key structural intersection results to settings where intersections are governed by homological (rather than purely combinatorial) complexity.

A plausible implication is that the methodology surrounding homological shatter functions will continue to inform advances in discrete geometry, particularly in the analysis and design of geometric transversals, covering numbers, and quantitative intersection theorems on complex topological spaces. Furthermore, results on the slow-growth regime illuminate the minimal topological assumptions required for strong intersection properties, guiding the development of new Helly-type theorems and structural insights in combinatorial topology.

References:

• "Intersection patterns of set systems on manifolds with slowly growing homological shatter functions" (Avvakumov et al., 6 Jan 2026)
• "A fractional Helly theorem for set systems with slowly growing homological shatter function" (Bin, 2024)