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This paper proves that, among regular tournaments, the random tournament minimizes the density of a tournament H exactly when H is a transitive tournament or a blow-up of a cyclically oriented triangle with transitive parts.

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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.

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load-bearing objection A complete and essentially correct Sidorenko classification for regular tournamentons; the proof is sound and only needs small technical cleanups. the 2 major comments →

arxiv 2602.12551 v2 pith:XVR752A7 submitted 2026-02-13 math.CO

Sidorenko property and forcing in regular tournaments

Daniel Kr\'al' , Matja\v{z} Krnc , Filip Ku\v{c}er\'ak , Bernard Lidick\'y , Jan Volec This is my paper
classification math.CO MSC 05C2005C3505C80
keywords tournamentsSidorenko propertyquasirandom forcingtournament limitstournamentonsregular tournamentshomomorphism densityblow-ups of cyclic triangle
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 reading

Tournaments are complete directed graphs. A tournament has the Sidorenko property if, among all large tournaments whose vertices are all balanced (out-degree about half the number of vertices), the random tournament minimizes the number of copies of it. The paper characterizes this property completely: it holds exactly for transitive tournaments and for blow-ups of a cyclic triangle in which each part is transitive. As a consequence, it also characterizes which tournaments force quasirandomness in this regular setting, resolving two open problems about quasirandom forcing in nearly regular tournaments.

Core claim

The paper's central result, Corollary 10, is: for a tournament T, the inequality t(T,W) ≥ 2^{-binom(|V(T)|,2)} holds for every regular tournamenton W if and only if T is transitive or T is isomorphic to T[a,b,c] for natural numbers a,b,c — the tournament obtained from the cyclically oriented triangle by replacing each vertex with a transitive tournament. Corollary 11 sharpens this: the constant tournamenton W≡1/2 is the unique minimizer of t(T,W) among regular tournamentons precisely when T is transitive with at least four vertices or T[a,b,c] with a+b+c≥4. Together these give a complete description of the Sidorenko property and of quasirandom-forcing tournaments in the class of nearly regul

What carries the argument

Tournament limits are represented by measurable functions W:[0,1]^2→[0,1] with W(x,y)+W(y,x)=1, called tournamentons; regular tournamentons model nearly regular sequences. The paper's positive direction uses three specially designed digraphs: the blow-up C[a,b,c] of a cyclic triangle with independent twin parts, its sibling T[a,b,c] with transitive parts, and the auxiliary digraph B[c] (a single source directing c paths to a single sink). For regular W, the identity t(C[1,1,c],W)=t(B[c],W)/2 reduces the problem to a Jensen/Hölder inequality for B[c], which is then lifted to all C[a,b,c] via Lemma 6 and Hölder's inequality, and finally to T[a,b,c] via Lemma 8 (adding a transitive tournament t

Load-bearing premise

The entire negative direction of the characterization rests on Lemma 3, which asserts that every non-transitive tournament that contains none of the three small tournaments W4, L4 and C5 is necessarily a blow-up of the cyclic triangle; if that dichotomy is false, the list in the main theorem is incomplete.

What would settle it

Run an exhaustive search over small tournaments (up to 8 or 9 vertices) that are non-transitive but contain neither W4, L4 nor C5; if any of them is not isomorphic to some T[a,b,c], Lemma 3 (and the 'only if' direction of Corollary 10) fails. Alternatively, construct a regular tournamenton W and a tournament T outside the listed families with t(T,W) < 2^{-binom(|V(T)|,2)}.

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

If this is right

  • There are infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments: all T[a,b,c] with a+b+c≥4.
  • Almost every tournament is not quasirandom forcing for nearly regular tournaments, since forcing tournaments must lie in a sparse list and any anti-Sidorenko candidate has at most nine vertices.
  • The classification of the Sidorenko property among regular tournaments is exact; no other tournament has the random tournament as a density minimizer over the regular class.
  • The random tournament is the unique minimizer among regular tournamentons precisely for transitive tournaments with at least four vertices and for T[a,b,c] with a+b+c≥4, completing the forcing list on the minimizer side.

Where Pith is reading between the lines

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

  • The structural dichotomy in Lemma 3, if correct, suggests a general phenomenon: requiring a Sidorenko-type inequality in a degree-regular setting forces the extremal objects into iterated blow-up forms, which may guide conjectures for other regular limits such as regular permutation or hypergraph limits.
  • The entropy-method proof sketched for the case c=1 could, if completed for all parameters, provide a conceptual information-theoretic explanation of Theorem 7 and possibly extend the inequality to other tripartite blow-ups beyond the cyclic triangle.
  • A finite exhaustive search over tournaments with up to nine vertices could settle the remaining open case: which of them are maximizers among regular tournamentons, thereby completing the full classification of quasirandom forcing tournaments in the regular setting.
  • The strong difference between the global forcing list (only two tournaments up to isomorphism) and the regular forcing list (infinitely many) suggests that imposing degree regularity dramatically enriches the forcing phenomenon; quantifying this gap for other structures could be a fruitful direction.
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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 / 5 minor

Summary. The paper characterizes, in the language of tournamentons, the tournaments H whose homomorphism density among regular (equivalently, nearly regular) tournaments is minimized by the constant tournamenton. The main result, Corollary 10, states that t(H,W) ≥ 2^{-binom(|V(H)|,2)} for every regular tournamenton W if and only if H is a transitive tournament or one of the tournaments T[a,b,c], i.e., blow-ups of the cyclic triangle whose parts induce transitive tournaments. The proof is built from a structural dichotomy (Lemma 3) saying that the only non-transitive tournaments avoiding W4, L4 and C5 are the T[a,b,c]; a construction of H-free regular tournamentons for all other tournaments (Theorem 4); analytic inequalities proving the Sidorenko-type lower bound and equality case for T[a,b,c] (Lemmas 5–7, Theorems 7 and 9); and a twin-blowing inequality (Lemma 8). Corollary 11 identifies the quasirandom-forcing tournaments among these, answering two questions of Noel, Ranganathan and Simbaqueba and reducing their Problem 6.1 to a finite check via Proposition 12.

Significance. The result, if correct, is a complete and striking classification: unlike the graph setting, the Sidorenko property over regular tournamentons has a very small family of extremal tournaments. It resolves two open problems and provides a clean equality case. The proof is unusually transparent: the key inequalities (10), (15)–(17) are explicit and checkable, and the argument relies only on standard external facts (transitive tournament densities, the P4 quasirandomness criterion, compactness of tournamentons). The twin-blowing lemma is a nice tool. The paper is honest about where its method does not extend (the entropy-method sketch in Section 5) and labels the computational finding as an aside. I see no circularity and no hidden free parameters.

major comments (2)
  1. [Section 4, definition before Lemma 6; Eqs. (4), (12), (15)] D_{W,k} is defined as a quotient with denominator N^+_{W,k} N^-_{W,k}, but this denominator vanishes on a non-trivial subset of [0,1]^k for many tournamentons. Later identities and the proof of Lemma 6 treat the product N^+ D N^- as 0 on that set, but the convention is never stated. Please define D_{W,k} (or the product) explicitly on the null set so that (4), (12), and (15) are pointwise meaningful.
  2. [Section 4, Theorem 7 proof, inequality (15)] The generalized Hölder step sets p_2=ab/(ab-b) and p_3=ab/(b-a), which are undefined or infinite when a=1 or a=b. These cases are included in the theorem (e.g., T[1,2,c] and T[a,a,c]), so the proof as written does not cover the full statement. A short limiting argument for the infinite-exponent cases, or a separate treatment, is needed. The statement of generalized Hölder in Section 2 should also be adjusted to allow this convention.
minor comments (5)
  1. [Section 3, Lemma 3, last paragraph] The sentence "every triple of vertices of V_i induces a transitive tournament (otherwise, the triple and the vertex u_{i+1} would form W4)" appears to name the wrong tournament: with the established orientation of edges from V_i to V_{i+1}, u_{i+1} would be a sink, hence L4, not W4. Since both W4 and L4 are forbidden, the conclusion is unaffected. Also, "for every i∈V_i" should read "for every i∈[3]."
  2. [Section 2, generalized Hölder inequality] The statement says p_1,...,p_k are non-negative reals, but reciprocals 1/p_i appear. Please state p_i>0 and allow p_i=∞ with the usual convention, or restrict to the finite-exponent cases actually needed after the limiting argument in Theorem 7.
  3. [Section 5, Proposition 12] Regularity of the constructed tournamenton is asserted implicitly by the displayed definition but not verified. It is a short integral check and should be included. The inequality 2(2n)^{-n} > 2^{-binom(n,2)} for n≥10 should also be justified in one line.
  4. [Section 5, paragraph before Proposition 12] The sentence "every quasirandom forcing tournament has either the Sidorenko property or the anti-Sidorenko property" is used to reduce Problem 1 to finitely many cases, but no proof or reference is given. Please add a reference or a short argument; as written, this is an unsupported step in the secondary claim about Problem 6.1.
  5. [Section 5, computational aside] The claim that three 6-vertex tournaments were "computationally verified" to have the constant tournamenton as the unique maximizer among regular tournamentons is not reproducible without code or a description of the computation. Since this is an aside, it could be documented or moved to an appendix; otherwise it should be labelled as an unverified computational remark.

Circularity Check

0 steps flagged

No circularity in the central derivation; the classification is proven from stated inequalities and external inputs, with only non-load-bearing background self-citations.

full rationale

The central classification in Corollary 10 is not circular. The negative direction is a genuine constructive reduction: Lemma 3 is a standalone structural dichotomy (proved by a case analysis relative to a cyclic triangle) and Theorem 4 then exhibits explicit regular tournamentons — carousel tournaments for W4/L4 and iterated cyclic blow-ups for C5 — in which the forbidden tournaments have density zero; any tournament outside the listed class contains one of these by Lemma 3. The positive direction is a chain of analytic inequalities. Lemma 5 derives the B[k] lower bound from Jensen and the external P4 criterion [11]; Lemma 6 proves the C[a,b,k] inequality by Jensen; Theorem 7 combines these with generalized Hölder and the identity t(C[1,1,c],W)=t(B[c],W)/2, which is explicitly proved from regularity of W; Lemma 8 proves the transitive-blow-up step by induction. None of these steps fits a parameter to the target density or assumes the Sidorenko/forcing conclusion. The only self-citations (e.g. [31] in the introduction and preliminaries) are background or references to the tournamenton formalism, not load-bearing; the quasirandom/transitive-density inputs are external ([11], [21], [38]). Technical issues — the infinite exponent in generalized Hölder when a=b, the undefined D_k off its support, and the uncoded computational side remark in Section 5 — are limiting technicalities, not circularity; the entropy-method note is explicitly a non-generalization and is not used in the proof.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim is a pure existence/minimization theorem with no fitted constants. The paper defines auxiliary digraphs B[k], C[a,b,c], T[a,b,c] and the functions N+, N-, D_k for the proof; these are mathematical definitions, not new physical or conceptual entities. All nontrivial background results are cited to external prior work.

axioms (6)
  • standard math Generalized Hölder's inequality (stated at the end of Section 2)
    Used in Theorem 7, eq. (15), to bound the product of three integrals; supplied with proof by the authors.
  • standard math Jensen's inequality for convex functions
    Used in Lemma 5 (Jensen on G^k) and Lemma 6 (Jensen over probability measures defined by N+ and N-).
  • standard math Proposition 1: transitive tournament density and equality cases
    Cited to [21] and [38]; used in Corollaries 10 and 11 to handle transitive tournaments.
  • standard math Chung–Graham P4 characterization: F(x,y)=1/4 a.e. implies W≡1/2 for regular tournamentons
    Cited to [11]; used in Lemma 5 to identify the equality case of t(B[k],W).
  • domain assumption Compactness and limit theory of tournamentons
    Regular tournamentons represent limits of nearly regular tournament sequences; cited to [31,40]. Bridges the discrete statements to the analytic proofs.
  • standard math Entropy identities and conditional entropy facts in Section 5 sketch
    Used only in the discussion of an alternative proof for c=1; not part of the main theorem.

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Cite this review

Pith. "Pith review of Sidorenko property and forcing in regular tournaments." pith.science (2026). https://pith.science/paper/XVR752A7

@misc{pith2026260212551,
  author       = {Pith},
  title        = {Pith review of: Sidorenko property and forcing in regular tournaments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XVR752A7}},
  note         = {Machine review of arXiv:2602.12551}
}
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read the original abstract

We give a complete characterization of tournaments H that have the Sidorenko property with respect to nearly regular tournaments, i.e., the homomorphism density of H among all nearly regular tournaments is minimized by a random tournament. Corollaries of our result are a positive answer to the question of Noel, Ranganathan and Simbaqueba whether there exist infinitely many non-transitive tournaments that are quasirandom forcing for nearly regular tournaments, and a negative answer to their question whether almost every tournament is quasirandom forcing for nearly regular tournaments.

Figures

Figures reproduced from arXiv: 2602.12551 by Bernard Lidick\'y, Daniel Kr\'al', Filip Ku\v{c}er\'ak, Jan Volec, Matja\v{z} Krnc.

Figure 1
Figure 1. Figure 1: The unique quasirandom forcing tournament that is not transitive. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The digraphs S +[3], S +[4], S −[3] and S −[4]. C[2, 1, 1] C[3, 1, 1] C[2, 2, 1] [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The digraphs C[2, 1, 1], C[3, 1, 1] and C[2, 2, 1], which are blow-ups of the cyclically oriented triangle with parts of the sizes given by the parameters. T[2, 1, 1] T[3, 1, 1] T[2, 2, 1] [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The tournaments T[2, 1, 1], T[3, 1, 1] and T[2, 2, 1]. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The digraphs B[2], B[3] and B[4]. edges. The vertices u and v of a digraph are twins if the out-neighbors of u are exactly the out-neighbors of v, the in-neighbors of u are exactly the in-neighbors of v, and there is no edge between u and v. We write S +[k] and S −[k], where k ∈ N, for the orientation of the k-leaf star such that the central vertex is the source and the sink, respectively; see [PITH_FULL_… view at source ↗
Figure 6
Figure 6. Figure 6: The tournaments W4, L4 and C5. k be a positive integer. For any collection of measurable functions Fi : Ω → [0, 1], i ∈ [k], it holds that Z Ω Y i∈[k] Fi(x) dµ(x) ≤ Y i∈[k] Z Ω Fi(x) pi dµ(x) 1/pi whenever p1, . . . , pk are non-negative reals such that 1/p1 + · · · + 1/pk ≤ 1. 3 Constructions In this section, we present regular tournamentons witnessing that a tournament T does not have the Sidorenko pro… view at source ↗
Figure 7
Figure 7. Figure 7: The 5-vertex tournament induced by vertices [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The tournamenton W from the proof of Proposition 12 when T is the tournament W4 depicted in [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Three 6-vertex tournaments T that we have computationally verified to satisfy that the constant tournamenton is the unique maximizer of t(T, W) among regular tournamentons W. tournamenton is the unique maximizer of t(T, W) among regular tournamentons W; three 6-vertex tournaments T with this property are depicted in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.