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Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems

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disjoint-faces-in-simple-drawings-of-the-complete-graph-and-topologicalCombinatoricsmath.COposed by Alfredo Hubard, Andrew Sukrecorded: open · 1 machine check, unexamined

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Statement

The construction described above has Ω(n6)\Omega(n^6) crossings. Does there exist a cycle of small area in every drawing of KnK_n such that every pair of edges intersect a constant number of times?

Context

Candidate 1 of the open problems stated in "Disjoint faces in simple drawings of the complete graph and topological Heilbronn problems", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    Problem: Reconstructed statement: for every fixed constant CC there is a function εC(n)→0\varepsilon_C(n)\to0 such that every general-position drawing of KnK_n in [0,1]2[0,1]^2, with every two edges meeting in at most CC proper crossings, contains a nonzero graph cycle whose Z2\mathbb Z_2-inside has area at most εC(n)\varepsilon_C(n).
    This is the natural reading because Problem 5.4 appears inside the paper’s section on Z2\mathbb Z_2-cycles and follows the definition of Z2\mathbb Z_2-inside.

    Result: Yes. In fact, a 4-cycle suffices.

    Fix an edge abab. For each v∈V(Kn)∖{a,b}v\in V(K_n)\setminus\{a,b\}, let

    zv=ab+bv+va∈Z1(Kn;Z2),z_v=ab+bv+va\in Z_1(K_n;\mathbb Z_2),

    and let Rv⊂[0,1]2R_v\subset[0,1]^2 be its Z2\mathbb Z_2-inside.

    Because every pair of graph edges crosses at most CC times, each boundary ∂Rv\partial R_v has complexity bounded only by CC, and any two such boundaries meet in boundedly many points, again only depending on CC (the common edge abab is harmless, or may be separated by an arbitrarily small perturbation). Hence the family

    R={Rv:v≠a,b}\mathcal R=\{R_v:v\neq a,b\}

    has VC-dimension at most some constant d=d(C)d=d(C).

    By Haussler’s packing lemma, for normalized area measure μ\mu, any ε\varepsilon-separated subfamily under symmetric difference has size at most Kdε−dK_d\varepsilon^{-d}. Since ∣R∣=n−2|\mathcal R|=n-2, there exist distinct v,wv,w with

    μ(Rv△Rw)≤KC(n−2)−1/d.\mu(R_v\triangle R_w)\le K_C (n-2)^{-1/d}.

    But over Z2\mathbb Z_2,

    zv+zw=(ab+bv+va)+(ab+bw+wa)=av+vb+bw+wa,z_v+z_w=(ab+bv+va)+(ab+bw+wa)=av+vb+bw+wa,

    which is the 4-cycle avbwaa v b w a. Its Z2\mathbb Z_2-inside is exactly Rv△RwR_v\triangle R_w. Therefore this 4-cycle has area

    OC(n−1/d(C))=o(1).O_C(n^{-1/d(C)})=o(1).

    Thus every bounded-pairwise-intersection drawing of KnK_n contains a graph cycle of small Z2\mathbb Z_2-area.

    Citation: Uses the standard bounded-VC theorem for constant-complexity planar regions with bounded pairwise boundary intersections, and Haussler’s packing lemma: D. Haussler, “Sphere packing numbers for subsets of the Boolean nn-cube with bounded Vapnik–Chervonenkis dimension,” JCTA 69 (1995), 217–232.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The argument addresses the intended Z2Z_2-inside/o(1) version of Problem 5.4. For fixed intersection bound CC, the triangle interiors RvR_v have boundary/arrangement complexity bounded in terms of CC, so the family has bounded VC dimension. Haussler’s packing lemma then gives two Rv,RwR_v,R_w with o(1)o(1) symmetric-difference area. Their mod-2 boundary sum cancels the common edge abab and is exactly the 4-cycle avbwaa v b w a, whose Z2Z_2-inside is Rv△RwR_v\triangle R_w. Thus a small-area cycle exists.

      Novelty assessment

      TYPE1

      Classification rationale: The argument appears genuinely new as an explicit resolution of Hubard–Suk Problem 5.4, but it is a very short and fairly routine application of standard VC-dimension/packing machinery. Once one fixes an edge abab, the Z2Z_2-interiors of the triangles abvabv form a bounded-complexity range family; Haussler’s packing lemma immediately gives two with small symmetric difference, hence a small-area 4-cycle. This is a neat observation, but probably too minor for a standalone paper except perhaps as a short note.

      Literature check: I searched the arXiv, the published Discrete & Computational Geometry version, Semantic Scholar, Springer, and general web/Yahoo/Bing-indexed sources for combinations of “topological Heilbronn”, “Z2Z_2-cycles in topological graphs”, “cycle of small area”, “every pair of edges intersect a constant number of times”, and the exact Problem 5.4 wording. I found only the original Hubard–Suk paper and its published version; no later paper, note, forum post, or survey appears to state this VC-packing resolution. Semantic Scholar lists essentially only the original/published records and no evident solving paper.

      Citation: Hubard, A.; Suk, A. “Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems,” Discrete Comput. Geom. 74 (2025), 899–916, Problem 5.4. Uses Haussler, D. “Sphere packing numbers for subsets of the Boolean nn-cube with bounded Vapnik–Chervonenkis dimension,” JCTA 69 (1995), 217–232.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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