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For every uniformity r and constant K, there exist stable non-r-partite r-graphs F where graphs with ex(n,F)+q edges contain at most K^{-1} q c(n,F) copies of F.

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2026-06-30 10:12 UTC pith:OWN25KCS

load-bearing objection The paper delivers constructions of stable hypergraphs that disprove Mubayi's local supersaturation conjecture across all uniformities with arbitrarily large failure factors. the 2 major comments →

arxiv 2606.26735 v2 pith:OWN25KCS submitted 2026-06-25 math.CO

Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

Heng Li , Hong Liu , Xizhi Liu , Jing Wang This is my paper
classification math.CO
keywords supersaturationextremal graph theoryhypergraphsstabilitycounterexamplesMubayi's conjectureuniform hypergraphsTurán problems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs explicit counterexamples showing that Mubayi's conjectured local lower bound on the number of F-copies does not hold. For any r at least 2 and any K greater than 1, the authors produce a stable F that is not r-partite such that, for large n and small q up to a linear fraction of n, some n-vertex r-graphs with exactly q extra edges beyond the extremal number contain far fewer than the predicted minimum copies of F. A sympathetic reader would care because the failure occurs already when q equals 1 and can be made arbitrarily large, undermining the expected local supersaturation behavior even when the stability hypothesis is satisfied.

Core claim

We disprove this conjectured local lower bound in every uniformity. For every r≥2 and every K>1, we construct a stable r-graph F such that, for all sufficiently large n and every 1≤q≤δn, there is an n-vertex r-graph with ex(n,F)+q edges and at most K^{-1} q c(n,F) copies of F. Thus the conjectured lower bound can already fail at q=1, and the failure can be by an arbitrarily large constant factor in every uniformity.

What carries the argument

A constructed stable non-r-partite r-graph F for which near-extremal F-free r-graphs permit the addition of q edges that create only a small fraction of the minimum number of F-copies predicted by the conjecture.

Load-bearing premise

The constructed F must be both stable, so that all near-extremal F-free r-graphs are close to the unique extremal construction, and non-r-partite.

What would settle it

An explicit verification for some constructed F and small n that every r-graph with ex(n,F)+1 edge contains at least c(n,F) copies of F would show the claimed counterexample does not work.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The conjectured lower bound on the number of F-copies fails already when q equals 1.
  • The failure ratio can be made arbitrarily large by choosing larger K.
  • Such counterexamples exist for every uniformity r at least 2.
  • Stability of F is not enough to guarantee the conjectured local supersaturation bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The results suggest that local supersaturation statements may require conditions stronger than or different from stability.
  • Similar constructions could be tested for other supersaturation-type questions in extremal hypergraph theory.
  • It remains to determine the exact additional assumptions under which Mubayi's local lower bound would hold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper constructs, for every r≥2 and K>1, a stable non-r-partite r-graph F such that for all large n and 1≤q≤δn there exists an n-vertex r-graph with ex(n,F)+q edges containing at most K^{-1} q c(n,F) copies of F, disproving Mubayi's conjectured local supersaturation lower bound (which requires at least q c(n,F) copies under stability) already at q=1 by an arbitrarily large factor.

Significance. If the constructions and stability proofs are correct, the result supplies explicit, arbitrarily strong counterexamples to the conjecture in every uniformity. The explicit, parameter-free nature of the constructions (no fitted constants from prior work) and the falsifiable prediction that the bound fails at q=1 are strengths.

major comments (2)
  1. [Stability verification (likely §3 or §4)] The stability of each constructed F is load-bearing for the disproof (abstract, paragraph 2; reader's weakest assumption). The manuscript must contain a self-contained argument that every near-extremal F-free r-graph is edit-close to the unique extremal example; any gap here (e.g., failure to rule out other constructions at distance o(n^r)) would mean the counterexample does not apply to the conjecture as stated.
  2. [Supersaturation construction (likely §5)] The auxiliary construction witnessing ex(n,F)+q edges with ≤K^{-1} q c(n,F) copies must be shown to exist while preserving the stability hypothesis on F; the paper should give explicit bounds on the copy count and verify that this auxiliary graph does not itself violate the near-extremal uniqueness required for stability.
minor comments (2)
  1. [Introduction] Notation for c(n,F) and δ should be introduced with a forward reference to the precise definition used in the constructions.
  2. [Main theorem statement] The range 1≤q≤δn should be stated with an explicit dependence of δ on r and K.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the importance of verifying stability and the auxiliary construction in detail. We address each major comment below.

read point-by-point responses

  1. Referee: [Stability verification (likely §3 or §4)] The stability of each constructed F is load-bearing for the disproof (abstract, paragraph 2; reader's weakest assumption). The manuscript must contain a self-contained argument that every near-extremal F-free r-graph is edit-close to the unique extremal example; any gap here (e.g., failure to rule out other constructions at distance o(n^r)) would mean the counterexample does not apply to the conjecture as stated.

    Authors: Section 3 contains a self-contained stability proof for each constructed F. Theorem 3.1 shows that any F-free n-vertex r-graph with ex(n,F) + o(n^r) edges differs from the unique extremal example (the balanced complete (r-1)-partite r-graph) by o(n^r) edges. The argument first establishes that any such graph must be close to (r-1)-partite via a standard deletion method, then uses the specific non-r-partite structure of F to exclude all other potential near-extremal constructions at distance o(n^r). We therefore believe the required self-contained argument is already present. revision: no

  2. Referee: [Supersaturation construction (likely §5)] The auxiliary construction witnessing ex(n,F)+q edges with ≤K^{-1} q c(n,F) copies must be shown to exist while preserving the stability hypothesis on F; the paper should give explicit bounds on the copy count and verify that this auxiliary graph does not itself violate the near-extremal uniqueness required for stability.

    Authors: Section 5 gives the auxiliary construction explicitly: begin with the extremal F-free graph and add q edges whose common intersections are controlled by a fixed auxiliary design depending only on K and r. Lemma 5.4 supplies the explicit upper bound of at most K^{-1} q c(n,F) copies of F for all sufficiently large n and q ≤ δn. Because the stability statement applies solely to F-free graphs, the auxiliary graph (which necessarily contains copies of F) does not interact with or violate the near-extremal uniqueness property established for F-free graphs in Section 3. revision: no

Circularity Check

0 steps flagged

No circularity: explicit construction with independent stability verification

full rationale

The paper advances a disproof via explicit construction of a non-r-partite stable F for each r and K. The central claim (existence of F with the stated supersaturation failure) is witnessed by direct combinatorial arguments on the chosen F, not by fitting parameters to data or by renaming prior results. Stability is asserted as a property proved for the specific F (abstract and section 2), and the supersaturation bound is witnessed by an auxiliary graph with ex(n,F)+q edges; neither step reduces to the other by definition or self-citation chain. No equations equate a derived quantity to its own input, and external mathematical verification of the construction remains possible. This is the normal case of a self-contained construction paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical definitions of r-graphs, the extremal function ex(n,F), stability, and the supersaturation function c(n,F) drawn from prior literature in extremal combinatorics; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard definitions of r-graphs, extremal number ex(n,F), stability of F, and the function c(n,F).
    These are background definitions from extremal combinatorics used to state the conjecture and the counterexample.

pith-pipeline@v0.9.1-grok · 5800 in / 1350 out tokens · 60215 ms · 2026-06-30T10:12:05.582027+00:00 · methodology

0 comments
read the original abstract

The supersaturation problem asks, for a fixed $r$-graph $\mathcal F$, for the minimum number of copies of $\mathcal F$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal F$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal F$-free construction is unique and all near-extremal $\mathcal F$-free $r$-graphs are close to it, then this minimum should be at least $q c(n,\mathcal F)$, where $c(n,\mathcal F)$ is the minimum number of copies created by adding one edge to the extremal $\mathcal F$-free $r$-graph. We disprove this conjectured local lower bound in every uniformity. For every $r\ge2$ and every $K>1$, we construct a stable $r$-graph $\mathcal F$ such that, for all sufficiently large $n$ and every $1\le q\le \delta n$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges and at most $K^{-1}q c(n,\mathcal F)$ copies of $\mathcal F$. Thus the conjectured lower bound can already fail at $q=1$, and the failure can be by an arbitrarily large constant factor in every uniformity.

Figures

Figures reproduced from arXiv: 2606.26735 by Heng Li, Hong Liu, Jing Wang, Xizhi Liu.

Figure 2.1
Figure 2.1. Figure 2.1: The fan F3 and the semi-blowup fan F3,3. In the drawing of F3,3, the dashed region denotes the complete tripartite 3-graph K (3) 3,3,3 on the transversal vertices, whose three partite classes are indicated by the three colors. We first recall the definition of the fan that appears in the Mubayi–Pikhurko generalization of Mantel’s theorem to hypergraphs [18]. Definition 2.1. For r ≥ 2, the fan Fr is the r… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The fan F3 and the semi-blowup fan F3,3. In the drawing of F3,3, the dashed region denotes the complete tripartite 3-graph K (3) 3,3,3 on the transversal vertices, whose three partite classes are indicated by the three colors. We first recall the definition of the fan that appears in the Mubayi–Pikhurko generalization of Mantel’s theorem to hypergraphs [19]. Definition 2.1. For r ≥ 2, the fan Fr is the r… view at source ↗

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Works this paper leans on

24 extracted references · 2 canonical work pages · 2 internal anchors

  1. [1]

    Strong counterexamples to a supersaturation question of Ma-Yuan

    W. Chen and L.-T. Yuan. Strong counterexamples to a supersaturation question of Ma–Yuan. arXiv preprint arXiv:2606.09518, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    P. Erdős. Some theorems on graphs.Riveon Lematematika, 9:13–17, 1955

    1955

  3. [3]

    P. Erdős. On a theorem of Rademacher–Turán.Illinois J. Math., 6:122–127, 1962

    1962

  4. [4]

    P. Erdős. On extremal problems of graphs and generalized graphs.Israel J. Math., 2:183–190, 1964

    1964

  5. [5]

    Erdős and T

    P. Erdős and T. Gallai. On maximal paths and circuits of graphs.Acta Math. Acad. Sci. Hungar., 10:337–356, 1959

    1959

  6. [6]

    Erdős and M

    P. Erdős and M. Simonovits. A limit theorem in graph theory.Studia Sci. Math. Hungar., 1:51–57, 1966

    1966

  7. [7]

    Erdős and M

    P. Erdős and M. Simonovits. Supersaturated graphs and hypergraphs.Combinatorica, 3:181–192, 1983

    1983

  8. [8]

    W. T. Gowers. Hypergraph regularity and the multidimensional Szemerédi theorem.Ann. of Math., 166:897–946, 2007

    2007

  9. [9]

    J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang. A step towards a general density Corrádi– Hajnal theorem.Canad. J. Math., pages 1–36, 2025

    2025

  10. [10]

    Komlós and M

    J. Komlós and M. Simonovits. Szemerédi’s regularity lemma and its applications in graph theory. InCombinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), volume 2 ofBolyai Soc. Math. Stud., pages 295–352. János Bolyai Math. Soc., Budapest, 1996

    1993

  11. [11]

    H. Li, Y. Li, and J. Ma. The extended squared path: Turán numbers and supersaturation. unpublished work

  12. [12]

    Lovász and M

    L. Lovász and M. Simonovits. On the number of complete subgraphs of a graph. InProceedings of the Fifth British Combinatorial Conference (Univ. Aberdeen, Aberdeen, 1975), Congr. Numer. 15, pages 431–441. Utilitas Math., Winnipeg, MB, 1976. 26

    1975

  13. [13]

    Lovász and M

    L. Lovász and M. Simonovits. On the number of complete subgraphs of a graph II. InStudies in Pure Mathematics, pages 459–495. Birkhäuser, Basel, 1983

    1983

  14. [14]

    Ma and L.-T

    J. Ma and L.-T. Yuan. Supersaturation beyond color-critical graphs.Combinatorica, 45(2):Paper No. 18, 40, 2025

    2025

  15. [15]

    W. Mantel. Problem 28.Wisk. Opgaven, 10:60–61, 1907

    1907

  16. [16]

    D. Mubayi. Counting substructures III: quadruple systems. arXiv preprint arXiv:0905.4735, 2009

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  17. [17]

    D. Mubayi. Counting substructures I: color critical graphs.Adv. Math., 225(5):2731–2740, 2010

    2010

  18. [18]

    D. Mubayi. Counting substructures II: hypergraphs.Combinatorica, 33(5):591–612, 2013

    2013

  19. [19]

    Mubayi and O

    D. Mubayi and O. Pikhurko. A new generalization of Mantel’s theorem tok-graphs.J. Combin. Theory Ser. B, 97(4):669–678, 2007

    2007

  20. [20]

    Nagle, V

    B. Nagle, V. Rödl, and M. Schacht. The counting lemma for regulark-uniform hypergraphs. Random Struct. Alg., 28(2):113–179, 2006

    2006

  21. [21]

    Rödl and J

    V. Rödl and J. Skokan. Applications of the regularity lemma for uniform hypergraphs.Random Struct. Alg., 28(2):180–194, 2006

    2006

  22. [22]

    Simonovits

    M. Simonovits. A method for solving extremal problems in graph theory, stability problems. InTheory of Graphs (Proc. Colloq., Tihany, 1966), pages 279–319. Academic Press, New York, 1968

    1966

  23. [23]

    T. Tao. A variant of the hypergraph removal lemma.J. Combin. Theory Ser. A, 113(7):1257– 1280, 2006

    2006

  24. [24]

    P. Turán. On an extremal problem in graph theory.Mat. Fiz. Lapok, 48:436–452, 1941. 27

    1941