Forcing Quasirandomness in a Regular Tournament

Arjun Ranganathan; Jonathan A. Noel; Lina M. Simbaqueba

REVIEW 3 major objections 4 minor 1 cited by

Forcing Quasirandomness in a Regular Tournament

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Among the twenty tournaments on at most five vertices, exactly eleven force quasirandomness once the hosts are required to be nearly regular; the other nine do not.

desk verdict A sensible new variant of quasirandomness forcing with a complete 5-vertex classification; the positive half rests on flag-algebra certificates that need machine-checkable backing. read the letter →

arxiv 2501.11675 v2 pith:SDT5ES27 submitted 2025-01-20 math.CO

Jonathan A. Noel , Arjun Ranganathan , Lina M. Simbaqueba This is my paper · ORCID

classification math.CO MSC 05C2005C35

keywords quasirandomnesstournamentsnear-regularityforcingpropertyhomomorphismdensityflagalgebratournamentlimitsclassification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Random tournaments look the same at every scale: every finite pattern appears with its random probability, and a sequence of tournaments with these frequencies is called quasirandom. The paper asks which small patterns H can certify quasirandomness by themselves: if the frequency of H in a large tournament matches its random expectation, must all other pattern frequencies match too? In the unrestricted setting, only the transitive tournaments on four or more vertices and one exceptional five-vertex tournament have this property. The authors add a mild near-regularity assumption, that almost every vertex has out-degree close to half the number of vertices, and give the full picture for tournaments on at most five vertices: eleven force quasirandomness under this assumption and nine do not. The point is that the forcing family becomes substantially larger, so there are many more single-pattern tests for quasirandomness once near-regularity is known.

What carries the argument

The central objects are tournamentons, measurable functions $W:[0,1]^2 \to [0,1]$ with $W(x,y)+W(y,x)=1$, which are the continuous limits of tournament sequences; quasirandomness corresponds to $W(x,y)=1/2$ almost everywhere. The key reduction, Proposition 3.8, is that near-regularity is exactly the condition that the transitive triangle density or the cyclic triangle density sits at its random value, so H forces quasirandomness in regular tournaments precisely when the set {H, TT3} or {H, C3} forces it without regularity. On the positive side, the proof uses the flag algebra method, a calculus of rooted subpattern densities that turns density inequalities into positive semidefinite matrix certificates; the printed matrices A1, A2 and A3 yield inequalities such as $8t(C_3,W) + (1/4)1024t(H_{10},W) \le 5/4$, and the equality analysis forces the two four-vertex densities $t(TT_4,W)$ and $t(C_4,W)$ to equal $(1/8)t(TT_3,W)$, which Lemma 5.13 converts into $W=1/2$ almost everywhere. On the negative side, Lemma 3.10 builds regular tournamentons by splicing two regular tournamentons and uses continuity to hit the random density while remaining non-quasirandom.

What would settle it

Recompute the eigenvalues and kernels of the printed matrices $A_1,A_2,A_3$ and re-evaluate, for every five-vertex tournament $J$, the claimed constant expressions in Theorems 5.4–5.7; a matrix that is not positive semidefinite, a kernel different from the stated span, or a tournament $J$ whose expression differs from $5/4$, $8/7$ or $6/5$ would refute the positive half of the classification. An independent exhaustive search over regular tournamentons on a fine grid could look for a regular limit $W$ with t(H_i,W) equal to the random value for one of the eleven listed tournaments while W is not $1/2$ on a positive-measure set.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is the exact split of the twenty tournaments: H4, H5, H6, H7, H8, H10, H11, H13, H14, H15 and H17 force quasirandomness in regular tournaments, while H0, H1, H2, H3, H9, H12, H16, H18 and H19 do not. The positive half is proved by reducing regular forcing to ordinary forcing of a pair: by Proposition 3.8, H forces quasirandomness in regular tournaments if and only if {H, TT3}, equivalently {H, C3}, forces quasirandomness without the regularity assumption. The flag algebra inequalities of Theorems 5.4–5.7 then certify the needed pairs for H10, H11, H13 and H14, with H15 obtained by reversing arcs, and equality is shown to force the limit tournamenton to be $W(x,y)=1/2$ for almost all pairs. The negative half is witnessed by explicit regular tournamentons, including one-parameter interpolations between two regular limits and a regular seven-vertex tournament, for which t(H,W) reaches the random value while W stays far from the constant $1/2$.

Load-bearing premise

The load-bearing premise is that the printed positive semidefinite matrices $A_1,A_2,A_3$ and the Appendix A coefficient tables are all correct; the paper gives no code or certificates, and a single wrong entry would break Theorems 5.4–5.7 and with them the positive half of the classification.

Editorial extensions

If this is right

In any nearly regular tournament sequence, matching the expected density of any one of the eleven listed tournaments forces every finite pattern density to match its random value.
The forcing list strictly enlarges the unrestricted one on five vertices: H5, H6, H7, H10, H11, H13, H14 and H15 are non-transitive tournaments that force only under near-regularity, while H4, H8 and H17 remain forcing without it.
Because near-regularity is equivalent to t(TT3) or t(C3) being at the random value, the classification can be restated as finitely many inequalities involving three- and five-vertex densities, all with equality only at the constant $1/2$ limit.
Any future resolution of the open classification problem must reproduce this exact split on five vertices; the paper's Questions 6.2 and 6.3 ask whether the family continues to infinity and whether it eventually contains almost all tournaments.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

This inference goes beyond the paper: the negative constructions all live on low-complexity regular limits built from small regular tournaments or interpolations between them, so the non-forcing phenomenon may be confined to a finite-dimensional slice of the space of tournamentons.
This inference goes beyond the paper: the same flag algebra pipeline could be run on the 112 six-vertex tournaments to test Question 6.3 at the next scale; the paper's method stops at five vertices.
This inference goes beyond the paper: for tournament data with approximately balanced outdegrees, matching any one of the eleven pattern counts is a practical quasirandomness certificate, provided near-regularity is checked separately.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper does something genuinely useful. It weakens quasirandomness forcing by assuming the host tournaments are nearly regular, and then fully resolves the case of at most five vertices. The winning half—eleven forcing tournaments—contrasts sharply with the unrestricted case, where only transitive tournaments of size at least 4 and H17 force. The definition is natural, and the main theorem is the obvious next step after the work of Bucić–Long–Shapira–Sudakov and Hancock et al.

What is good: the structural reductions in Section 3 are clean. Proposition 3.8, which equates forcing in regular tournaments with forcing in the presence of TT3 or C3, is a neat observation that makes the whole enterprise tractable. The negative constructions are mostly explicit rational tournamentons; the H9/H16 example is a straightforward hand computation, and the H6 proof via [4, Corollary 6] is elegant. The flag algebra part is standard but carefully laid out, with the equality-case arguments driven by kernel conditions rather than by fitting the matrices to the target densities.

The soft spot is the positive half of Theorem 1.4. The PSD assertions for the 8x8 matrices A2 and A3 (and A1 in Theorem 5.7) are stated without proof or certificate, and the identity for each 5-vertex tournament J is only checked for one or two sample J, with the rest delegated to the appendix. The appendix appears complete for the twelve 5-vertex tournaments, but the coefficient tables are themselves computer-generated and not independently machine-checked. This is a verification gap, not evidence of error—the sample arithmetic is consistent and the method is routine—but it is exactly the thing a referee should ask the authors to back up with an exact certificate, such as rational eigenvalues or floating-point SDP output in a reproducible form. A minor follow-on: the H18 negative construction uses an approximate numerical value for t(H18, WT); the margin over 1/1024 is large, so I am not worried, but a rational lower bound would tie it off cleanly.

Who this is for: people working on quasirandomness, flag algebras, or tournament limits. It does not resolve the general classification or connect outside the field, but it opens a concrete new question—are almost all tournaments forcing under near-regularity?—and it provides a small complete case with a mix of methods.

Recommendation: send it to a serious referee. The main theorem is plausible and the proof architecture is sound; the missing certificates are fixable and should be the main request. I would not desk reject.

Referee Report

3 major / 4 minor

Summary. The paper introduces and studies a variant of quasirandom forcing for tournaments in which the host tournaments are assumed to be nearly regular. The main result, Theorem 1.4, classifies all tournaments on at most five vertices that force quasirandomness in regular tournaments: exactly eleven do (H4, H5, H6, H7, H8, H10, H11, H13, H14, H15, H17) and exactly nine do not (H0, H1, H2, H3, H9, H12, H16, H18, H19). The negative side is handled by explicit regular tournamenton constructions, including a neat hand computation for H9 and H16 and a computer-assisted example for H18. The positive side is proved by reducing the problem to inequalities on homomorphism densities of C3 or TT3 combined with H10, H11, H13, or H14, and then proving these inequalities with flag algebras. The paper also gives structural reductions (Propositions 3.1–3.10) that connect near regularity, density of TT3 and C3, and forcing in regular tournaments, and it closes with several natural open problems, including whether almost every tournament forces quasirandomness in regular tournaments.

Significance. If the central results are correct, this paper significantly enlarges the known family of quasirandom-forcing tournaments by adding a regularity assumption on the host sequence. This is a meaningful conceptual contrast to Theorem 1.2, where only one non-transitive tournament forces quasirandomness. The structural reductions are clean and appear sound, and the negative constructions are explicit and mostly hand-checkable. The positive half, however, relies on flag-algebra certificates that are only partially verified in the text: large matrices are asserted to be positive semidefinite, kernel claims are asserted without proof, and the coefficient tables in Appendix A are stated as computer-generated with only sample demonstrations. These are load-bearing for the main classification, but the issue is a verification gap rather than a detected mathematical error. The paper does not ship machine-checkable code or certificates, which would substantially strengthen confidence in the positive results.

major comments (3)

[Theorems 5.4–5.7 and Lemma 5.18]
[Proofs of Theorems 5.4–5.7, equality cases]
[Lemma 4.4, H18 construction]

minor comments (4)

[Proof of Theorem 5.5]
[Appendix A]
[Theorem 5.7]
[Proposition 4.3]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the flag-algebra certificates are asserted rather than fitted, and the equality cases are derived from kernel conditions.

full rationale

The derivation chain is not circular. Proposition 3.8 converts near-regular forcing to forcing with TT3 or C3, and Lemma 3.9 gives the reduction used in Section 5; both are proved from Propositions 3.6/3.7, not assumed. The negative results in Section 4 are explicit tournamenton constructions (WC3, the Uz family, and a computer-searched 7-vertex tournament for H18), so the non-forcing conclusions are exhibited rather than presupposed. The positive half rests on Lemma 5.18: the matrices A1, A2, A3 and the Appendix A coefficient tables are asserted to give, for every 5-vertex J, the constant value (-5/4, -5/4, 8/7, 6/5). The paper only shows sample evaluations and says the tables were 'computed by computer but can be easily checked by hand'; this is an unverified-certificate risk, not circularity, because the PSD matrices are certificates chosen to prove the inequalities and are not fitted to the densities t(H,1/2) that define the target value. The equality cases are then obtained from the kernel assertions in Lemma 5.18 together with Lemma 5.13, whose proof uses the external Theorem 1.2. The one self-citation that is load-bearing is Lemma 5.2 = [4, Corollary 6] for H6; it is an external, parameter-free inequality about C3/C4 densities with assumptions that do not include the present classification, so it counts as independent evidence and does not raise the circularity score. Overall, the paper's claims reduce neither to their own definitions nor to a fitted parameter renamed as a prediction.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim leans on unverified computer-generated inputs: flag algebra PSD matrices, kernel descriptions, Appendix A coefficient tables, and a 7-vertex tournament found by exhaustive search. The text calls these computer-checkable, but no code or certificate is shipped. Everything else is standard combinatorics or proved in the paper.

free parameters (6)

PSD matrix A2 in Theorem 5.4 = 8x8 matrix with entries printed in Section 5
Chosen by computer to certify the helper inequality; PSD and kernel asserted without proof.
PSD matrix A3 in Theorem 5.4 = 8x8 matrix printed in Section 5
Chosen by computer to certify the helper inequality; PSD and kernel asserted without proof.
PSD matrices A2 and A3 in Theorem 5.5 = two 8x8 matrices printed in Section 5
Chosen by computer to certify the helper inequality; PSD and kernel asserted without proof.
PSD matrix A2 in Theorem 5.6 = 8x8 matrix printed in Section 5
Chosen by computer to certify the helper inequality; PSD and kernel asserted without proof.
PSD matrices A1 and A3 in Theorem 5.7 = 4x4 and 8x8 matrices printed in Section 5
Chosen by computer to certify the helper inequality; PSD and kernel asserted without proof.
7-vertex regular tournament T in Lemma 4.4 = 7x7 adjacency matrix printed in proof of Lemma 4.4
Found by exhaustive computer search; provides the required t(H18, W_T) > 1/1024 example; no code supplied.

assumptions (5)

standard math Every sequence of tournaments has a subsequence converging to a tournamenton, and convergence preserves homomorphism densities (Proposition 2.1).
Used as the bridge from finite tournaments to limits in Sections 2-5.
domain assumption Near-regularity of a sequence is equivalent to the limit tournamenton being regular, expressed through t(TT3) and t(C3) limits (Propositions 3.6 and 3.7).
Proved in text from Proposition 3.1, but it is the modeling assumption that defines the paper's problem variant.
standard math The flag algebra Lemma 5.18 is valid: positive semidefinite flag matrices yield valid lower bounds on homomorphism densities, with the given equality condition.
The paper proves the lemma from the density decomposition; it is part of the standard Razborov framework.
ad hoc to paper The printed matrices A1, A2, A3 in Theorems 5.4-5.7 are positive semidefinite and have exactly the stated kernels.
Asserted without proof or certificate; all equality-case arguments in the positive half depend on this.
ad hoc to paper The coefficient tables in Appendix A are complete and correct for the twelve 5-vertex tournaments, and the 7-vertex tournament in Lemma 4.4 has the stated density.
Reported as computer-computed, with no code or certificates; the negative result for H18 and all flag algebra inequalities rely on them.

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Pith. "Pith review of Forcing Quasirandomness in a Regular Tournament." pith.science (2026). https://pith.science/paper/SDT5ES27

@misc{pith2026250111675,
  author       = {Pith},
  title        = {Pith review of: Forcing Quasirandomness in a Regular Tournament},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDT5ES27}},
  note         = {Machine review of arXiv:2501.11675}
}

abstract

A tournament $H$ is said to force quasirandomness if it has the property that a sequence $(T_n)_{n\in \mathbb{N}}$ of tournaments of increasing orders is quasirandom if and only if the homomorphism density of $H$ in $T_n$ tends to $(1/2)^{\binom{v(H)}{2}}$ as $n\to\infty$. It was recently shown that there is only one non-transitive tournament with this property. This is in contrast to the analogous problem for graphs, where there are numerous graphs that are known to force quasirandomness and the well known Forcing Conjecture suggests that there are many more. To obtain a richer family of characterizations of quasirandomness in tournaments, we propose a variant in which the tournaments $(T_n)_{n\in \mathbb{N}}$ are assumed to be "nearly regular." We characterize the tournaments on at most 5 vertices which force quasirandomness under this stronger assumption.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Sidorenko property and forcing in regular tournaments
math.CO 2026-02 conditional novelty 7.0 of 10

For nearly regular tournaments, a tournament has the Sidorenko property exactly when it is transitive or a blow-up of the cyclic triangle whose three parts are transitive.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Alon and J

N. Alon and J. H. Spencer. The probabilistic method . Wiley Series in Discrete Mathe- matics and Optimization. John Wiley & Sons, Inc., Hoboken, NJ, fourt h edition, 2016

work page 2016

[2] [2]

Buci´ c, E

M. Buci´ c, E. Long, A. Shapira, and B. Sudakov. Tournament q uasirandomness from local counting. Combinatorica, 41(2):175–208, 2021

work page 2021

[3] [3]

Burke, B

D. Burke, B. Lidick´ y, F. Pfender, and M. Philips. Inducibility of 4- vertex tournaments. E-print arXiv:2103.07047v2, 2022

arXiv 2022

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T. F. N. Chan, A. Grzesik, D. Kr´ al’, and J. A. Noel. Cycles of lengt h three and four in tournaments. J. Combin. Theory Ser. A , 175:105276, 23, 2020

work page 2020

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T. F. N. Chan, D. Kr´ al’, J. A. Noel, Y. Pehova, M. Sharifzadeh, and J. Volec. Character- ization of quasirandom permutations by a pattern sum. Random Structures Algorithms, 57(4):920–939, 2020

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F. Chung. From quasirandom graphs to graph limits and graphlets . Adv. in Appl. Math., 56:135–174, 2014

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