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Overlap-Helly theorems

Andreas F. Holmsen, Alfredo Hubard

arXiv:2609.10023v1math.COcs.CGmath.GT

TL;DR

The paper asks how classical Helly-type theorems can be viewed through overlap results, where some fiber meets many faces or sets. It proves a colorful overlap theorem for continuous maps and convex families, and uses this framework to derive further Helly-type results.

  • Problem

    The paper asks whether classical Helly-type theorems, including fractional Helly, admit overlap formulations.

    Source: §1 · p. 1

  • Method

    It studies continuous maps into d-dimensional simplicial complexes and proves an overlap analogue of colorful Helly for families of convex sets.

    Source: §1 · p. 1

  • Results

    If every transversal choice of d+1 sets from the d+1 convex families intersects, some fiber of every continuous map to a d-dimensional complex meets every member of one family.

    Source: §1 · p. 1

  • Takeaways & Limitations

    The framework relates classical Helly-type results to overlap theorems, and an affirmative answer to Problem 21 would make each overlap Helly theorem follow from its d-Leray counterpart.

    Source: §1 · p. 1, §4 · p. 15

    The authors omit details for one argument.

    Source: §3 · p. 14

Abstract

from arXiv · show

In this paper we introduce a generalization of Helly's theorem closely connected to B\'ar\'any-Gromov overlap theorems (also called selection lemmas). Our main result implies both the topological colorful Helly of Kalai and Meschulam and Karasev's topological centerpoint theorem. We further investigate the topological fractional Helly theorem from this overlap perspective, and show an overlap theorem for dense complexes (a continuous second selection lemma for tame maps).

1. Introduction

The paper treats overlap theorems as geometric-topological analogues of the pigeonhole principle and uses them to recast classical Helly-type results. Its main theorem gives a common fiber meeting every set in one of several families under a colorful intersection condition.

Main overlap theorem

  • Under the colorful intersection condition, every continuous map to a d-dimensional simplicial complex has a fiber meeting every member of one family.The families consist of compact convex sets in R^m, with m > d, and every transversal choice of d+1 sets intersects.Source: §1 · p. 1

Topological Helly consequences

  • Theorem 1 yields colorful Helly for good covers, while Corollary 4 replaces good-cover assumptions with dimension-dependent connectivity conditions.The good-cover result follows through a convex-set construction whose nerve is the chosen subcomplex and whose sets map into the cover members.Source: §1 · p. 2

Centerpoint connection

  • Karasev’s topological centerpoint theorem follows from Corollary 2 when the target is triangulable.For facewise linear maps into R^d, the stated topological centerpoint theorem reduces to the centerpoint theorem.Source: §1 · pp. 1–3

Sliced-cube theorem

  • For maps from a subdivided (d+1)-cube to a d-dimensional complex, one fiber meets every slice in one coordinate direction.The unit-cube version gives a fiber meeting every slice in that direction, and the paper also states a waist-type consequence.Source: §1 · pp. 3–4

Further overlap questions

  • The overlap version of the weak ε-net theorem states that, for each integer d≥1 and ε∈(0,1), some p=p(ε,d) makes every continuous f: Δ_N→R^d admit p points whose preimages collectively intersect every face of dimension ⌈εN⌉.Source: §1 · p. 4

Fractional Helly

  • If continuous maps from a complex K have the s-overlap property in dimension d, convex families with nerve K satisfy overlap fractional Helly with β ≥ s/(d+1).The resulting fiber meets at least β|F| members of the family.Source: §1 · p. 5

Dense and random complexes

  • Dense complexes and random complexes provide special cases of overlap results for continuous maps.For an α-dense complex, the paper gives a (1−a)μ_d-overlap guarantee when α > 1−aμ_d; for the specified random complex, the μ_d-overlap property holds with probability tending to one.Source: §1 · p. 5

Tame maps

  • For sufficiently dense d-complexes, ℓ-tame maps to a d-manifold satisfy an overlap property with a positive bound depending on α, d, and ℓ.The theorem requires at least an α-fraction of the d-faces and guarantees some s ≥ s0(α,d,ℓ).Source: §1 · pp. 5–6

2. Overlap for the sliced cube

The proof establishes that a continuous map of the sliced cube has a fiber meeting every slice in one fixed coordinate direction. A balanced-complex path lemma supplies the simplicial case, and approximation and compactness extend it to continuous maps.

2. Overlap for the sliced cube

  • For a simplicial map from a triangulated (d+1)-cube to a d-dimensional balanced complex, some vertex fiber contains a path joining disjoint cube facets.A monochromatic path from the HEX lemma maps to a constant vertex because adjacent vertices of a balanced complex cannot share a color and form an edge.Source: §2 · p. 6
  • The proof extends the simplicial result to continuous maps by approximating the map with a simplicial map into a subdivision of the target complex.The subdivision is balanced, and the approximation preserves the slice images up to an arbitrarily small neighborhood.Source: §2 · pp. 6–7
  • A continuous-map fiber meets every slice S_i,s for all s in [0,1], with one fixed coordinate index i.The proof obtains the continuous statement by rescaling finer cube slices and passing to a compact limit.Source: §2 · p. 7
  • Compactness turns fibers meeting every slice in a sequence of finer grids into a limiting fiber meeting every slice.A convergent subsequence gives a fixed coordinate direction, a limiting target point, and convergence of the compact fibers in the Hausdorff metric.Source: §2 · p. 7

3. Overlap and the fractional Helly theorem

This section connects overlap properties of simplicial complexes to fractional Helly and weak epsilon-net results, then proves positive overlap for dense complexes under tame maps. It also derives overlap for sufficiently dense random subcomplexes.

3. Overlap and the fractional Helly theorem

  • The nerve construction transfers face-overlap in K to a fiber that intersects members of the convex family F.The map sends each face into the union of its corresponding convex sets; the overlap fiber then yields the family-intersection conclusion.Source: §3 · pp. 10–11
  • At most s^-1 epsilon^-(d+1) points are needed in the greedy selection procedure for the weak epsilon-net result.Each newly selected point covers at least an s·epsilon^(d+1) fraction of previously uncovered simplices.Source: §3 · p. 11
  • With probability tending to one, sufficiently dense random subcomplexes satisfy the expansion and cohomology conditions used to obtain a positive overlap fraction.The argument applies the topological overlap theorem after establishing cofilling bounds and local sparsity.Source: §3 · pp. 11–12
  • A dense d-skeleton yields positive overlap for ℓ-tame maps from K to R^d.The proof combines a dense-subcomplex count with disjoint simplices whose images share crossing points, then counts incidences to find a fiber meeting a positive fraction of d-faces.Source: §3 · pp. 12–14
  • A more quantitatively precise Erdős–Simonovits bound is said to improve the lower bound on s, but its derivation is omitted.The stated refinement is not worked out in the paper passage.Source: §3 · p. 14

4. Concluding remarks

The concluding remarks relate overlap approaches to both d-Leray and bounded-Betti-number generalizations, while identifying representation and interpolation questions that would connect overlap theorems to d-Leray Helly theorems.

4. Concluding remarks

  • The authors frame overlap approaches as relevant to both d-Leray and bounded-Betti-number generalizations of topological Helly theorems.They note that earlier approaches used different strategies for these two settings.Source: §4 · p. 14
  • The remarks identify matroidal colorings and arbitrary d-Leray ambient complexes as two aspects of Kalai and Meshulam’s topological colorful Helly theorem not covered by their theorem.The authors describe extending their result to these aspects as open directions.Source: §4 · p. 15
  • For a d-dimensional complex Y with vanishing d-dimensional homology, the nerve of any good cover in Y is d-Leray.The proof uses the Nerve Theorem and the vanishing of homology in dimensions at least d for the relevant subcomplexes.Source: §4 · p. 15
  • In dimension 1, 1-Leray complexes are exactly clique complexes of chordal graphs and can be represented as intersection graphs of subtrees of a tree.The text states this as the converse to the preceding claim for d = 1.Source: §4 · p. 15
  • The remarks propose representing any d-Leray complex as the nerve of a good cover in a d-dimensional complex with vanishing d-dimensional homology.This is posed as Problem 20.Source: §4 · p. 15
  • If Problem 21 has an affirmative answer, every overlap Helly theorem follows directly from its corresponding Helly theorem for d-Leray complexes.Problem 21 asks whether a d-Leray complex L can be found with K ⊂ L ⊂ M.Source: §4 · p. 15
  • A subsequent paper reports a positive answer to Problem 10 and derives consequences.The remarks do not specify those consequences here.Source: §4 · p. 15
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