Intersection patterns in spaces with a forbidden homological minor
Pith reviewed 2026-05-24 12:56 UTC · model grok-4.3
The pith
Families avoiding a homological minor K as intersection pattern have fractional Helly number at most μ(K)+1.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For every simplicial complex K and integer b, every (K,b)-free cover has fractional Helly number at most μ(K)+1; consequently the (p,q)-theorem holds for every p ≥ q > μ(K) and every such cover.
What carries the argument
(K,b)-free cover, which requires K to be a forbidden homological minor of the ambient space while every nonempty subcollection intersection has reduced Betti numbers less than b in dimensions 0 through dim(K)−1; the proof proceeds by Ramsey arguments plus stair convexity in an auxiliary cubical complex.
Load-bearing premise
Violating the fractional Helly bound must produce a copy of K as a homological minor inside a cubical complex built from the intersections.
What would settle it
An explicit (K,b)-free cover whose intersection graph or hypergraph requires more than μ(K)+1 sets to guarantee a point in the common intersection.
read the original abstract
In this paper we study generalizations of classical results on intersection patterns of set systems in $\mathbb{R}^d$, such as the fractional Helly theorem or the $(p,q)$-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor. Given a simplicial complex $K$ and an integer $b$, we say that a family $\mathcal{F}$ of subcomplexes of some simplicial complex $X$ is a $(K,b)$-free cover if (i) $K$ is a forbidden homological minor of $X$, and (ii) the $j$th reduced Betti number $\tilde{\beta}_j(\bigcap_{S\in {\mathcal{G}}}S,\mathbb{Z}_2)$ is strictly less than $b$ for all $0\leq j < \dim K$ and all nonempty subfamilies $\mathcal{G}\subseteq \mathcal{F}$. We show that for every $K$ and $b$, the fractional Helly number of a $(K,b)$-free cover is at most $\mu(K)+1$, where $\mu(K)$ is the maximum sum of the dimensions of two disjoint faces in $K$. This implies that the assertion of the $(p,q)$-theorem holds for every $p \ge q > \mu(K)$ and every $(K,b)$-free cover $\mathcal{F}$. For $b=1$ and a suitable $K$ this recovers the original $(p,q)$-theorem and its generalization to good covers. Interestingly, our results show that that the range of parameters $(p,q)$ for which the $(p,q)$-theorem holds is independent of $b$. Our proofs use Ramsey-type arguments combined with the notion of stair convexity of Bukh et al. to construct (forbidden) homological minors in certain cubical complexes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper generalizes the fractional Helly theorem and (p,q)-theorem to triangulable spaces X with a forbidden homological minor K. A family F of subcomplexes is a (K,b)-free cover if K is forbidden as a homological minor of X and every nonempty intersection subfamily has reduced Betti numbers <b in dimensions 0 to dim(K)-1. The central theorem states that any such cover has fractional Helly number at most μ(K)+1, where μ(K) is the maximum sum of dimensions of two disjoint faces of K; this yields the (p,q)-theorem for all p≥q>μ(K). The range of valid (p,q) is independent of b. Proofs combine Ramsey arguments with stair convexity to construct homological minors inside auxiliary cubical complexes. For b=1 and suitable K the result recovers the classical (p,q)-theorem for good covers in R^d.
Significance. If the central claim holds, the work supplies a clean topological unification of Helly-type theorems, showing that the fractional Helly number is controlled solely by the combinatorial parameter μ(K) of the forbidden minor and is insensitive to the Betti bound b. This recovers and extends several known results in a uniform way and isolates the role of the forbidden minor from homological complexity of the intersections.
major comments (1)
- [Proof of the fractional Helly bound (via Ramsey + stair convexity)] The contradiction argument (abstract and the section describing the proof) assumes a violating intersection pattern, applies Ramsey-type arguments plus stair convexity to obtain K as a homological minor inside an auxiliary cubical complex, and claims this contradicts the hypothesis that K is forbidden in X. No explicit embedding, projection, or transfer map is described that would force the auxiliary minor to appear as a homological minor of the original space X itself, nor is it shown that the Betti-number restrictions enforced on subcomplexes of X propagate to the auxiliary construction. Because this step is load-bearing for the central claim, the gap must be closed.
minor comments (2)
- The abstract states that the result recovers the classical (p,q)-theorem 'for a suitable K'; an explicit description or reference to that K (e.g., the boundary of a simplex of dimension d) should appear in the introduction or a preliminary section.
- Notation μ(K) is introduced without a worked example; adding a short paragraph or figure illustrating μ(K) for K equal to a cycle or a complete bipartite graph would improve readability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying the load-bearing step in the proof of the fractional Helly bound. The comment correctly notes that the manuscript does not supply an explicit transfer mechanism between the auxiliary cubical complex and the original space X. We will revise the relevant section to close this gap.
read point-by-point responses
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Referee: [Proof of the fractional Helly bound (via Ramsey + stair convexity)] The contradiction argument (abstract and the section describing the proof) assumes a violating intersection pattern, applies Ramsey-type arguments plus stair convexity to obtain K as a homological minor inside an auxiliary cubical complex, and claims this contradicts the hypothesis that K is forbidden in X. No explicit embedding, projection, or transfer map is described that would force the auxiliary minor to appear as a homological minor of the original space X itself, nor is it shown that the Betti-number restrictions enforced on subcomplexes of X propagate to the auxiliary construction. Because this step is load-bearing for the central claim, the gap must be closed.
Authors: We agree that the current write-up leaves the transfer step implicit. The auxiliary cubical complex is constructed directly from the intersection pattern of the given (K,b)-free cover inside X by taking products of simplices corresponding to the stair-convex chains; the Ramsey argument then produces a homological minor of K inside this cubical complex. Because the cubical complex is assembled from subcomplexes of X whose Betti numbers are already bounded by hypothesis, the homology classes realizing the minor can be pushed forward along the natural inclusion maps into X, yielding a homological minor of K in X itself. We will add a short lemma (and accompanying diagram) that makes this push-forward explicit and verifies that the dimension and Betti-number constraints are preserved. This revision will be placed immediately after the stair-convexity construction in the proof of the fractional Helly theorem. revision: yes
Circularity Check
No circularity: derivation relies on external Ramsey arguments and Bukh et al. stair convexity
full rationale
The paper states a theorem deriving the fractional Helly bound μ(K)+1 for (K,b)-free covers from Ramsey-type arguments plus the external stair-convexity construction of Bukh et al. to produce homological minors in auxiliary cubical complexes. No equations, definitions, or steps reduce the claimed bound to a fitted quantity or to the input assumptions by construction. No self-citation is invoked as load-bearing justification for a uniqueness theorem or ansatz. The result is presented as a self-contained theorem whose proof chain does not collapse to its own inputs; any questions about transfer from the auxiliary complex concern proof validity rather than circular reduction.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Reduced Betti numbers are well-defined invariants of simplicial complexes over Z_2
- domain assumption Existence of stair convexity and its interaction with cubical complexes to produce homological minors
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We prove Theorem 4 using ... stair convexity of Bukh et al. [6] offers a systematic way of building chain maps from simplicial complexes into grid-like complexes (Proposition 8).
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discussion (0)
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