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Hypergraph Turán Problems in ℓ_2-Norm

Combinatorics · math.CO · posed by József Balogh, Felix Christian Clemen, Bernard Lidický · open

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Statement

π(K5=)=227387andσ(K5=)=29098127.\pi(K_5^=) = \frac{227}{387} \quad \text{and} \quad \sigma(K_5^=) = \frac{2909}{8127}.

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  • Hypergraph Turán Problems in ℓ_2-Norm
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: For a 3-uniform hypergraph HH, let

    π(H)=lim⁡n→∞ex⁡(n,H)(n3),σ(H)=lim⁡n→∞exco⁡2(n,H)(n2)(n−2)2.\pi(H)=\lim_{n\to\infty}\frac{\operatorname{ex}(n,H)}{\binom n3}, \qquad \sigma(H)=\lim_{n\to\infty} \frac{\operatorname{exco}_2(n,H)}{\binom n2(n-2)^2}.

    Here K5=K_5^= is the 3-graph on five vertices with eight edges, whose two missing triples intersect in exactly one vertex. The conjecture states

    π(K5=)=227387,σ(K5=)=29098127.\pi(K_5^=)=\frac{227}{387},\qquad \sigma(K_5^=)=\frac{2909}{8127}.

    Result: The conjecture is false. In fact,

    π(K5=)≥3763>227387.\pi(K_5^=)\ge \frac{37}{63}>\frac{227}{387}.

    Let S6S_6 be the 3-graph on [6][6] with edges

    123,234,345,451,512,136,246,356,256,146.123,234,345,451,512,136,246,356,256,146.

    Every four vertices of S6S_6 span exactly two edges, so S6S_6 is K43−K_4^{3-}-free; hence every iterated balanced blow-up of S6S_6 is also K43−K_4^{3-}-free.

    Now take three equal vertex classes A,B,CA,B,C. Put in all triples of types

    ABC,AAB,BBC,CCA,ABC,\qquad AAB,\qquad BBC,\qquad CCA,

    and inside each of A,B,CA,B,C place an iterated balanced blow-up of S6S_6.

    This 3-graph is K5=K_5^=-free: any five-set with color pattern other than 4+14+1 in a class and its successor spans at most seven edges. In the exceptional 4+14+1 case it spans at most eight edges, and if it spans eight then the two missing triples lie inside the four-vertex class and therefore intersect in two vertices, giving K5<K_5^<, not K5=K_5^=.

    Its edge density is

    59+19⋅27=3763,\frac{5}{9}+\frac{1}{9}\cdot\frac{2}{7} =\frac{37}{63},

    because the top-level construction contributes 5/95/9, while the iterated S6S_6-blow-up has density 2/72/7 inside each of the three equal classes.

    Finally,

    3763−227387=22709>0.\frac{37}{63}-\frac{227}{387} =\frac{2}{2709}>0.

    Thus the asserted value π(K5=)=227/387\pi(K_5^=)=227/387 is impossible, so the conjunction in the conjecture is false.

    Citation: The construction uses the Frankl–Füredi S6S_6 iterated blow-up for K43−K_4^{3-}, as also described in Balogh–Clemen–Lidický, “Hypergraph Turán Problems in ℓ2\ell_2-Norm,” arXiv:2108.10406.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The construction is a valid K5=K_5^=-free sequence with density 37/6337/63. The key checks are sound: the iterated S6S_6-blow-up is K43−K_4^{3-}-free, so any four vertices inside one class span at most two edges; the finite color-pattern count gives at most seven edges on every five-set except the 4+14+1 successor case, where eight edges form K5<K_5^<, not K5=K_5^=. The density computation

    5/9+(1/9)(2/7)=37/635/9+(1/9)(2/7)=37/63

    is correct, and 37/63>227/38737/63>227/387. Thus the conjectured value of π(K5=)\pi(K_5^=) is disproved. I found no already-stated stronger literature result; the cited paper appears to contain the construction but undercounts the recursive S6S_6 contribution.

    Novelty assessment

    TYPE1

    Classification rationale: The construction gives a valid new lower bound π(K5=)≥37/63>227/387\pi(K_5^=)\ge 37/63>227/387, so it refutes the stated conjectural value. However it is a short construction/correction using the known Frankl–Füredi S6S_6 blow-up idea, and it does not determine the true Turán density or the codegree-squared density. On its own it looks more like a corrigendum or brief note than a standalone standard-journal paper.

    Literature check: I found no source stating the bound 37/6337/63, no source refuting Conjecture 3.6, and no later update of the Balogh–Clemen–Lidický survey incorporating this correction. Exact searches for the constants and object names, including “227/387227/387”, “2909/81272909/8127”, “K5=K_5^=”, and “37/6337/63” with hypergraph/Turán/codegree terms, did not reveal an existing reference. The original arXiv paper remains the relevant source for the conjecture and for the surrounding constructions.

    Citation: J. Balogh, F. C. Clemen, B. Lidický, “Hypergraph Turán Problems in ℓ2\ell_2-Norm,” arXiv:2108.10406, Conjecture 3.6. Also related: P. Frankl and Z. Füredi, “An exact result for 3-graphs,” Discrete Mathematics 50 (1984), for the K43−K_4^{3-}-free S6S_6 blow-up construction.

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