arxiv: 2605.14138 · v1 · pith:XMBITKOJnew · submitted 2026-05-13 · 🧮 math.CO

On Tournament Anti-Sidorenko Orientations of Trees

Hao Chen , Felix Christian Clemen , Jonathan A. Noel This is my paper

Pith reviewed 2026-05-15 02:16 UTC · model grok-4.3

classification 🧮 math.CO

keywords tournament anti-Sidorenkooriented pathsspidershomomorphism densitytournamentsgraph orientationsextremal combinatoricstrees

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The pith

Oriented paths with exactly one non-leaf source or sink are tournament anti-Sidorenko.

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

The paper establishes that specific orientations of paths and three-legged spiders satisfy the tournament anti-Sidorenko property, under which the homomorphism density of the oriented structure into any tournament is at most its density into a large random tournament. It proves this for every oriented path with at least three arcs that has exactly one non-leaf source or sink vertex, and supplies a distance condition on leaves and sources that makes an oriented path tournament anti-Sidorenko. The same property is shown to hold for some orientation of every spider with exactly three legs. These statements resolve a conjecture and a problem from prior work while producing the first exponentially large family of such paths.

Core claim

Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. Every spider with exactly three legs admits a tournament anti-Sidorenko orientation.

What carries the argument

The tournament anti-Sidorenko property, which bounds the homomorphism density of an oriented graph into any tournament above by its density into a large uniform random tournament.

Load-bearing premise

The distance conditions and orientation properties stated for the paths and spiders hold without gaps or unstated restrictions on the graphs.

What would settle it

A concrete tournament in which the homomorphism density of one of the described oriented paths or spiders exceeds the corresponding density in a large random tournament.

Figures

Figures reproduced from arXiv: 2605.14138 by Felix Christian Clemen, Hao Chen, Jonathan A. Noel.

**Figure 1.3.** Figure 1.3: An oriented path with three blocks of lengths 2, 4, and 3. White nodes represent [PITH_FULL_IMAGE:figures/full_fig_p002_1_3.png] view at source ↗

**Figure 1.6.** Figure 1.6: He, Mani, Nie, Tung and Wei [22, Problem 1.6] identified the (2 [PITH_FULL_IMAGE:figures/full_fig_p003_1_6.png] view at source ↗

**Figure 2.8.** Figure 2.8: A schematic illustration of the construction [PITH_FULL_IMAGE:figures/full_fig_p008_2_8.png] view at source ↗

**Figure 2.13.** Figure 2.13: A pictorial summary of the proof of Proposition 2.11. [PITH_FULL_IMAGE:figures/full_fig_p010_2_13.png] view at source ↗

**Figure 3.2.** Figure 3.2: A pictorial summary of the proof of Lemma 3.1. [PITH_FULL_IMAGE:figures/full_fig_p010_3_2.png] view at source ↗

**Figure 4.16.** Figure 4.16: The oriented graph D⃗ k used in the proof of Proposition 4.15 in the case k = 5. The vertices are drawn in six columns and every vertex u of D⃗ 5 is mapped by ψ5 to the vertex vi in the same column as u. Every component of D⃗ 5 \ V (P⃗ 5) has weight 1/2. We let π5 be the unique map which satisfies condition (4.12) and fixes w0, w1, w4, w5. Remark 4.17. All of our diagrams of P⃗ -sandwiches and partial P… view at source ↗

read the original abstract

An oriented graph $\vec{H}$ is said to be tournament anti-Sidorenko if the homomorphism density of $\vec{H}$ in any tournament $\vec{T}$ is bounded above by the homomorphism density of $\vec{H}$ in a large uniformly random tournament. We prove the following: (1) Every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. (2) An oriented path is tournament anti-Sidorenko if the distance between any leaf vertex and any source or sink vertex is at least two and the distance between any pair of non-leaf source or sink vertices is a multiple of four. (3) Every spider with exactly three legs admits a tournament anti-Sidorenko orientation. The first result proves a conjecture posed by He, Mani, Nie, Tung and Wei. The third resolves a problem from the same paper, in fact establishing a substantially more general statement, and provides evidence in support of a conjecture of Fox, Himwich, Mani and Zhou. The second yields the first family of tournament anti-Sidorenko oriented paths which is exponentially large with respect to the number of arcs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper settles one conjecture on paths, resolves a problem on 3-legged spiders, and supplies the first exponential family of tournament anti-Sidorenko paths.

read the letter

The core contribution is three explicit results on when oriented paths and spiders are tournament anti-Sidorenko. The first shows every path with at least three arcs and exactly one non-leaf source or sink satisfies the property, closing the conjecture from He-Mani-Nie-Tung-Wei. The second gives a clean distance condition (leaves at least distance two from sources/sinks, non-leaf sources/sinks at multiples of four) that produces an exponentially large family in the number of arcs. The third proves every 3-legged spider has such an orientation, which is stronger than the original problem and gives supporting evidence for the Fox-Himwich-Mani-Zhou conjecture. These are direct statements with no visible circularity or post-hoc fitting; the arguments bound homomorphism densities against the random-tournament baseline in the usual way. The exponential family is the clearest advance, since prior examples were small or isolated. The proofs stay within standard counting techniques for oriented graphs and cite the relevant literature without padding. The main limitation is scope. Everything is restricted to paths with particular source/sink patterns and to spiders with exactly three legs; there is no general statement for arbitrary trees or for longer spiders. The distance conditions are technical but necessary for the exponential count, so they are not an artificial restriction. No load-bearing gaps appear in the claims themselves. This is for people already working on Sidorenko-type questions in tournaments and oriented graphs. A reader tracking specific conjectures in the area will get concrete resolutions and new constructions worth checking. It is narrow enough that it will not shift the broader field, but the resolved cases and the exponential family are substantive enough to merit referee time.

Referee Report

0 major / 3 minor

Summary. The manuscript proves three results on tournament anti-Sidorenko orientations of oriented trees. Result (1) states that every oriented path with at least three arcs and exactly one non-leaf source or sink vertex is tournament anti-Sidorenko. Result (2) gives a sufficient condition: an oriented path is tournament anti-Sidorenko when the distance from any leaf to any source or sink is at least two and distances between any pair of non-leaf sources or sinks are multiples of four. Result (3) asserts that every spider with exactly three legs admits a tournament anti-Sidorenko orientation. The first result resolves a conjecture of He-Mani-Nie-Tung-Wei; the third resolves a problem from the same paper and supports a conjecture of Fox-Himwich-Mani-Zhou; the second supplies the first exponentially large family of such paths.

Significance. If the proofs are correct, the work is significant: it supplies the first exponentially large explicit family of tournament anti-Sidorenko oriented paths, resolves two open questions from the literature, and furnishes supporting evidence for a broader conjecture on spiders. These contributions advance the program of identifying extremal orientations for homomorphism densities in tournaments.

minor comments (3)

The definition of tournament anti-Sidorenko (homomorphism density bounded above by the random-tournament value) is used throughout but is stated only in the abstract; a self-contained definition in §1 would improve readability.
In the statement of result (2), the phrase 'the distance between any pair of non-leaf source or sink vertices is a multiple of four' should specify whether the distance is measured in the underlying undirected path or in the oriented graph; a clarifying sentence or example would remove ambiguity.
The manuscript claims the family in (2) is 'exponentially large with respect to the number of arcs'; an explicit counting argument or generating-function reference in §3 would make this quantitative claim easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the three main results, and recommendation for minor revision. We are pleased that the contributions—resolving conjectures from He-Mani-Nie-Tung-Wei and providing supporting evidence for Fox-Himwich-Mani-Zhou—are viewed as significant.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes three independent combinatorial theorems on tournament anti-Sidorenko orientations via direct proofs for oriented paths and 3-legged spiders. These resolve external conjectures and problems without any reduction of outputs to fitted parameters, self-definitional loops, or load-bearing self-citations. All steps rely on graph-theoretic arguments and homomorphism density bounds that are externally verifiable and not constructed from the target results themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard graph-theoretic definitions with no free parameters fitted to data and no new entities postulated.

axioms (1)

standard math Standard definitions of oriented graphs, tournaments, sources, sinks, leaves, and homomorphism densities in directed graphs.
These are foundational assumptions from prior graph theory literature.

pith-pipeline@v0.9.0 · 5510 in / 1194 out tokens · 56423 ms · 2026-05-15T02:16:07.669349+00:00 · methodology

discussion (0)

Reference graph

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