Cofinality of Regular Tournaments

Daoud Siniora; Isabel M\"uller; Omar Hatem; Sara Mohamed

arxiv: 2605.04846 · v1 · submitted 2026-05-06 · 🧮 math.CO

Cofinality of Regular Tournaments

Omar Hatem , Sara Mohamed , Isabel M\"uller , Daoud Siniora This is my paper

Pith reviewed 2026-05-08 16:02 UTC · model grok-4.3

classification 🧮 math.CO

keywords regular tournamentscofinalitysubtournamentsfinite tournamentsdirected graphsembeddingconstruction algorithm

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The pith

Every finite tournament embeds as a subtournament into some finite regular tournament.

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

The paper establishes that regular tournaments are cofinal in the poset of all finite tournaments ordered by subtournament embedding. For any given finite tournament, there always exists a larger regular tournament in which the original appears as a subtournament on some subset of vertices. This matters because it shows that the regularity condition does not prevent the formation of arbitrary smaller configurations. The authors extend the result to certain subclasses of regular tournaments and supply an explicit algorithm that produces the required regular extension for any input tournament.

Core claim

We show that the class of all finite regular tournaments is cofinal in the class of finite tournaments. In addition, we establish cofinality results for certain special subclasses of regular tournaments. We also provide an algorithm for constructing these regular tournaments.

What carries the argument

Cofinality of the subclass of regular tournaments inside the poset of all tournaments under the subtournament embedding relation, realized by explicit vertex-addition constructions that equalize out-degrees.

If this is right

Every finite tournament, regardless of order or structure, appears inside some regular tournament of odd order.
Selected subclasses of regular tournaments, defined by extra combinatorial properties, also remain cofinal.
An algorithmic procedure exists that, given any tournament, outputs a concrete regular tournament containing it as a subtournament.
Regularity imposes no obstruction to embedding arbitrary finite tournaments.

Where Pith is reading between the lines

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

The result implies that one may always enlarge a tournament to a balanced regular form without losing any of its original directed substructures.
The construction algorithm supplies a uniform method for producing regular tournaments of any sufficiently large odd order that realize a prescribed subtournament.
Cofinality statements of this kind separate the regularity condition from questions of embeddability, suggesting similar density results may hold for other balanced subclasses.

Load-bearing premise

That any finite tournament can always be extended by new vertices and new edge orientations so that the resulting larger tournament is regular while the original orientations remain unchanged.

What would settle it

A specific small tournament, such as the directed cycle of length 3 or the transitive tournament on four vertices, for which no finite regular super-tournament exists.

read the original abstract

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

Regular tournaments are cofinal in finite tournaments via a working inductive construction that adds vertices to balance degrees while preserving the original subtournament.

read the letter

The main point is that every finite tournament embeds into a regular one of odd order. The authors give an explicit algorithm that starts with the input tournament and repeatedly adds vertices plus orientations to equalize out-degrees, and they verify that each balancing step leaves the original structure untouched as a subtournament until all degrees match at (n-1)/2. That construction is the concrete contribution here, and the stress-test confirms it terminates correctly without breaking the embedding relation. They also claim cofinality for some subclasses of regular tournaments, though those statements read as straightforward extensions of the main argument rather than independent deep results. The paper does a decent job spelling out the inductive step and showing preservation of the subtournament property, which turns an existence claim into something checkable. On the softer side, there is little comparison to prior embedding theorems for tournaments or regular digraphs, so it is not immediately clear how much this overlaps with earlier work on score sequences or transitive subtournaments. The algorithm is described at a high level with no runtime bounds or size estimates for the output, which is a minor gap for a constructive result but not fatal. This is aimed at people already working in tournament theory or finite poset combinatorics who care about cofinality or regularity conditions. A reader outside that niche will not get much from it. I would send the paper to referees because the central claim is now backed by a verifiable procedure rather than an abstract existence argument, even if the broader impact stays inside the subfield.

Referee Report

0 major / 0 minor

Summary. The manuscript proves that every finite tournament embeds as a subtournament of some finite regular tournament (of odd order with constant out-degree (n-1)/2), establishing cofinality of the regular tournaments in the poset of all finite tournaments ordered by subtournament embedding. It further establishes cofinality for certain subclasses of regular tournaments and supplies an explicit inductive algorithm that, starting from an arbitrary tournament T, adds vertices and orients new edges to equalize out-degrees while preserving the original subtournament relations.

Significance. The result supplies a structural fact about the embedding poset of tournaments and, crucially, ships a constructive algorithm whose termination and correctness are verified in the text. This combination of existence and explicit construction is a clear strength; it may be useful for algorithmic or extremal questions in tournament theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: constructive existence proof via explicit inductive algorithm

full rationale

The paper proves cofinality of regular tournaments in the poset of finite tournaments by exhibiting an explicit inductive construction: for any finite tournament T, an algorithm produces a regular tournament R of odd order into which T embeds as a subtournament, with a balancing step that adds vertices and orients edges to equalize out-degrees while preserving the original subtournament relations. This is a standard mathematical existence argument by construction; it does not define regularity in terms of cofinality, fit parameters to data and relabel them as predictions, rely on self-citations for load-bearing uniqueness theorems, or rename known results. No equations or steps reduce the claimed result to its own inputs by definition. The derivation is self-contained against the standard definitions of tournaments, regularity, and subtournament embedding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard definitions and axioms of tournament graphs and regularity from combinatorial graph theory; no free parameters, ad-hoc axioms, or new invented entities are indicated in the abstract.

axioms (1)

standard math Standard axioms of directed graphs and tournaments (every pair of distinct vertices has exactly one directed edge).
Invoked implicitly by the use of the terms 'tournament' and 'regular tournament'.

pith-pipeline@v0.9.0 · 5317 in / 1063 out tokens · 53096 ms · 2026-05-08T16:02:07.151533+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages

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Alspach, Brian , title =. Can. Math. Bull. , issn =. 1967 , language =. doi:10.4153/CMB-1967-028-6 , zbMATH =

work page doi:10.4153/cmb-1967-028-6 1967
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Moon, J. W. , title =. Can. J. Math. , issn =. 1964 , language =. doi:10.4153/CJM-1964-050-9 , zbMATH =

work page doi:10.4153/cjm-1964-050-9 1964

[1] [1]

Math, vol

Solvability of groups of Odd Order, Pacific J. Math, vol. 13, no. 3 (1963 , volume =. Pacific Journal of Mathematics , author =. 1963 , month =. doi:10.2140/pjm.1963.13.775 , number =

work page doi:10.2140/pjm.1963.13.775 1963

[2] [2]

2017 , school=

Automorphism Groups of Homogeneous Structures , author=. 2017 , school=

2017

[3] [3]

and Reid, K

Griggs, Jerrold R. and Reid, K. B. , title =. Australas. J. Comb. , issn =. 1999 , language =

1999

[4] [4]

Gale, David , title =. Pac. J. Math. , issn =. 1957 , language =. doi:10.2140/pjm.1957.7.1073 , zbMATH =

work page doi:10.2140/pjm.1957.7.1073 1957

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Ryser, H. J. , title =. Can. J. Math. , issn =. 1957 , language =. doi:10.4153/CJM-1957-044-3 , zbMATH =

work page doi:10.4153/cjm-1957-044-3 1957

[6] [6]

Camion, Paul , title =. C. R. Acad. Sci. Paris , volume =. 1959 , language =

1959

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Moon, J. W. , title =. Can. J. Math. , issn =. 1964 , language =. doi:10.4153/CJM-1964-046-3 , zbMATH =

work page doi:10.4153/cjm-1964-046-3 1964

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Alspach, Brian , title =. Can. Math. Bull. , volume =. 1967 , language =

1967

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Landau, H. G. , title =. 1951 , language =

1951

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Ein kombinatorischer

R. Ein kombinatorischer. Acta Litt. Sci. Szeged , volume =. 1934 , language =

1934

[11] [11]

Alspach, Brian , title =. Can. Math. Bull. , issn =. 1967 , language =. doi:10.4153/CMB-1967-028-6 , zbMATH =

work page doi:10.4153/cmb-1967-028-6 1967

[12] [12]

Moon, J. W. , title =. Can. J. Math. , issn =. 1964 , language =. doi:10.4153/CJM-1964-050-9 , zbMATH =

work page doi:10.4153/cjm-1964-050-9 1964