Counterexamples to two conjectures about matroids

Matt Larson

arxiv: 2607.02208 · v1 · pith:IOPVJJ4Lnew · submitted 2026-07-02 · 🧮 math.CO

Counterexamples to two conjectures about matroids

Matt Larson This is my paper

Pith reviewed 2026-07-03 10:43 UTC · model grok-4.3

classification 🧮 math.CO MSC 05B35

keywords matroidscounterexamplesWhite's conjectureMason's conjecturetoric idealslog-concavityflats

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The pith

Counterexamples show that White's conjecture on toric ideals and Mason's conjecture on flat counts are false for matroids.

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

The paper supplies explicit matroids that serve as counterexamples to two established conjectures in the field. White's conjecture asserted that the toric ideal of any matroid is generated by the symmetric exchange binomials. Mason's conjecture asserted that the sequence counting the flats of each rank in a matroid is always log-concave. The constructions demonstrate that neither property holds in general. A reader would care because both conjectures had been used to predict uniform algebraic and combinatorial behavior across all matroids; their disproof shows that additional generators or non-log-concave sequences can occur.

Core claim

We give counterexamples to two well-known conjectures about matroids: White's conjecture on the generation of the toric ideal by symmetric exchange binomials, and a conjecture of Mason on the log-concavity of the counts of flats of a given rank.

What carries the argument

Explicitly constructed matroids that violate the conjectured generation property for toric ideals and the conjectured log-concavity of flat counts.

Load-bearing premise

The objects presented satisfy the axioms of a matroid and the direct checks that they violate the two conjectures are correct.

What would settle it

An independent verification that one of the constructed matroids is not a valid matroid or that its toric ideal is in fact generated by the symmetric exchange binomials and its flat counts are log-concave.

Figures

Figures reproduced from arXiv: 2607.02208 by Matt Larson.

**Figure 1.** Figure 1: Implications between some variants of White’s conjecture. The conjectures which are known to be false are colored red. Kashiwabara [Kas10]. Conjecture 1.2 was proven for split matroids in [BS24] and for regular matroids in [BMS26]. It was shown that the symmetric exchange binomials generate the toric ideal up to saturation in [LM14]. In [Las21], it was shown that the toric ideal of a matroid of rank r has … view at source ↗

**Figure 2.** Figure 2: The matroid in Example 2.2 A cyclic set of N with |S| − rkN (S) = 1 consists of either two elements of one of the large parallel classes or it intersects all three large parallel classes in one element and uses the edge in the path of length 1, giving W75 = 3 26 2 + 263 = 18, 551. A cyclic set of N with |S| − rkN (S) = 2 consists of either three elements of one large parallel class; two pairs from two l… view at source ↗

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

The paper supplies explicit counterexamples to White's toric ideal conjecture and Mason's log-concavity conjecture, but the abstract shows none of the actual constructions or checks.

read the letter

The main takeaway is that this paper claims to disprove two longstanding conjectures in matroid theory with concrete examples. White's conjecture concerns the toric ideal being generated by symmetric exchange binomials. Mason's conjecture concerns the log-concavity of the sequence that counts flats by rank.

What is new is the disproofs themselves. The paper does well by locating explicit matroids that violate both statements rather than offering only abstract arguments or reductions. That kind of concrete evidence can help the field move on to corrected or weaker versions of the conjectures.

The soft spot is verification. The abstract states the results but gives no ground sets, bases, circuits, or computations. The claims stand or fall on whether the constructed objects satisfy the matroid axioms and whether the ideal and flat-count checks are correct. If the paper presents small enough examples with explicit lists and shows the algebraic or enumerative calculations, the result is solid. If those steps are missing or opaque, the disproofs cannot be assessed. The citation pattern is standard and points back to the original statements of the conjectures.

This paper is for people working in matroid theory, especially those tracking algebraic and enumerative questions. A reader who follows these two conjectures would want to see the counterexamples to update their picture of what holds.

It deserves peer review so that referees can examine the constructions directly. I would send it out rather than desk reject.

Referee Report

3 major / 0 minor

Summary. The paper claims to construct explicit counterexamples to two conjectures in matroid theory: White's conjecture that the toric ideal associated to a matroid is generated by the symmetric exchange binomials, and Mason's conjecture that the sequence counting the number of flats of each rank is log-concave.

Significance. Disproofs of these two well-known conjectures would be significant for the field, as they would clarify the limitations of algebraic generation properties and combinatorial inequalities in matroids. Explicit constructions, if verified, provide concrete objects that can be studied further and may suggest refined statements or new questions about matroid ideals and flat lattices.

major comments (3)

The central claim rests on the validity of the constructed matroids and the explicit checks that they violate the conjectures (as noted in the weakest assumption). The manuscript must supply the ground sets, bases or circuits, and the algebraic/combinatorial verification steps for both examples so that independent confirmation of the independence axioms and the violations is possible; without these details the disproof cannot be assessed.
For the counterexample to White's conjecture, the paper should report the specific computation showing that the toric ideal requires additional generators beyond the symmetric exchange binomials (e.g., via Gröbner basis calculation or dimension comparison), including the ground-set size and the ideal generators used.
For the counterexample to Mason's conjecture, the paper should list the explicit counts of flats by rank for the constructed matroid and demonstrate the failure of log-concavity (e.g., by exhibiting three consecutive terms a, b, c with b² < a·c), together with confirmation that the object is a matroid.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential significance of explicit counterexamples to these conjectures. We address each of the major comments below. Where the referee correctly identifies the need for additional explicit data to permit independent verification, we have revised the manuscript accordingly.

read point-by-point responses

Referee: The central claim rests on the validity of the constructed matroids and the explicit checks that they violate the conjectures (as noted in the weakest assumption). The manuscript must supply the ground sets, bases or circuits, and the algebraic/combinatorial verification steps for both examples so that independent confirmation of the independence axioms and the violations is possible; without these details the disproof cannot be assessed.

Authors: We agree that the manuscript should contain sufficient explicit data for independent verification of both the matroid axioms and the claimed violations. In the revised version we have added the ground sets, complete lists of bases, and the verification steps confirming that the structures satisfy the independence axioms. We have also included the explicit algebraic and combinatorial checks demonstrating the violations of each conjecture. revision: yes
Referee: For the counterexample to White's conjecture, the paper should report the specific computation showing that the toric ideal requires additional generators beyond the symmetric exchange binomials (e.g., via Gröbner basis calculation or dimension comparison), including the ground-set size and the ideal generators used.

Authors: We have expanded the relevant section to include the requested computational details for the counterexample to White's conjecture. The revised manuscript now states the ground-set size, lists the symmetric exchange binomials, and reports the outcome of the Gröbner basis computation (or dimension comparison) establishing that additional generators are required. revision: yes
Referee: For the counterexample to Mason's conjecture, the paper should list the explicit counts of flats by rank for the constructed matroid and demonstrate the failure of log-concavity (e.g., by exhibiting three consecutive terms a, b, c with b² < a·c), together with confirmation that the object is a matroid.

Authors: The revised manuscript now contains a table of the explicit flat counts by rank for the matroid counterexample to Mason's conjecture. We exhibit three consecutive terms a, b, c satisfying b² < a·c to demonstrate the failure of log-concavity, and we include the basis list confirming that the object is a matroid. revision: yes

Circularity Check

0 steps flagged

No circularity: paper supplies explicit counterexamples via direct verification

full rationale

The paper claims to disprove two conjectures by constructing specific matroids and verifying that they violate the stated properties (toric ideal generation and log-concavity of flat counts). No derivation chain, fitted parameters, predictions, or uniqueness theorems are invoked; the load-bearing steps are axiom checks and explicit combinatorial/algebraic computations on the constructed objects. These are independent verifications, not reductions to the paper's own inputs by construction. Self-citations, if present, are irrelevant because the central claims do not rest on them. This is the standard non-circular case for a counterexample paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of a matroid and on the precise statements of the two conjectures from earlier papers; no new entities or fitted parameters are introduced in the abstract.

axioms (1)

standard math Standard matroid axioms (independence axioms or equivalent formulations)
The paper assumes its constructed objects satisfy the definition of a matroid.

pith-pipeline@v0.9.1-grok · 5542 in / 1039 out tokens · 43197 ms · 2026-07-03T10:43:10.583464+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 5 canonical work pages · 1 internal anchor

[1]

[AHK18] Karim Adiprasito, June Huh, and Eric Katz,Hodge theory for combinatorial geometries, Ann. of Math. (2)188 (2018), no. 2, 381–452. MR3862944 [AHL25] Omid Amini, June Huh, and Matt Larson,A decomposition theorem for Lefschetz modules(2025). arXiv:2511.02026v1. [Aig87] Martin Aigner,Whitney numbers, Combinatorial geometries, 1987, pp. 139–160. MR9210...

work page arXiv 2018
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arXiv:2606.13960v1

MR4737143 [BCL+26] Spencer Backman, Nathan Cheung, Micha l Laso´ n, Gaku Liu, and Mateusz Micha lek,The Gr¨ obner version of White’s conjecture is false(2026). arXiv:2606.13960v1. [BHM+20] Tom Braden, June Huh, Jacob Matherne, Nicholas Proudfoot, and Botong Wang,Singular Hodge theory for combinatorial geometries(2020). arXiv:2010.06088v4. [BL25] Spencer B...

work page arXiv 2026
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MR427110 [GL ¨O21] Cem G¨ uneri, San Ling, and Buket ¨Ozkaya,Quasi-cyclic codes, Concise encyclopedia of coding theory, 2021, pp. 129–149. MR4598739 [HH02] J¨ urgen Herzog and Takayuki Hibi,Discrete polymatroids, J. Algebraic Combin.16(2002), no. 3, 239–268 (2003). MR1957102 [HK12] June Huh and Eric Katz,Log-concavity of characteristic polynomials and the...

work page arXiv 2021
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(N.S.)27(2021), no

MR335518 [Las21] Micha l Laso´ n,On the toric ideals of matroids of a fixed rank, Selecta Math. (N.S.)27(2021), no. 2, Paper No. 18,

2021
[5]

Math.259(2014), 1–12

MR4236584 [LM14] Micha l Laso´ n and Mateusz Micha lek,On the toric ideal of a matroid, Adv. Math.259(2014), 1–12. MR3197649 [Mas72] J. H. Mason,Matroids: unimodal conjectures and Motzkin’s theorem, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), 1972, pp. 207–220. MR0349445 [McG20] Sean McGuinness,Frame matroids, toric ideals,...

2014
[6]

Simplex faces and quadratic toric ideals of lattice polytopes

MR4088420 [MO26] Aki Mori and Hidefumi Ohsugi,Simplex faces and quadratic toric ideals of lattice polytopes(2026). arXiv:2606.20430v1. [Rot71] Gian-Carlo Rota,Combinatorial theory, old and new, Actes du Congr` es International des Math´ ematiciens (Nice, 1970), Tome 3, 1971, pp. 229–233. MR505646 [Sch11] Jay Schweig,Toric ideals of lattice path matroids a...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[7]

White,A unique exchange property for bases, Linear Algebra Appl.31(1980), 81–91

MR1363949 [Whi80] Neil L. White,A unique exchange property for bases, Linear Algebra Appl.31(1980), 81–91. MR570381 [YY25] Yu-Chuan Yu and Chi Ho Yuen,White’s conjecture for paving matroids(2025). arXiv:2510.04163v1

work page arXiv 1980

[1] [1]

[AHK18] Karim Adiprasito, June Huh, and Eric Katz,Hodge theory for combinatorial geometries, Ann. of Math. (2)188 (2018), no. 2, 381–452. MR3862944 [AHL25] Omid Amini, June Huh, and Matt Larson,A decomposition theorem for Lefschetz modules(2025). arXiv:2511.02026v1. [Aig87] Martin Aigner,Whitney numbers, Combinatorial geometries, 1987, pp. 139–160. MR9210...

work page arXiv 2018

[2] [2]

arXiv:2606.13960v1

MR4737143 [BCL+26] Spencer Backman, Nathan Cheung, Micha l Laso´ n, Gaku Liu, and Mateusz Micha lek,The Gr¨ obner version of White’s conjecture is false(2026). arXiv:2606.13960v1. [BHM+20] Tom Braden, June Huh, Jacob Matherne, Nicholas Proudfoot, and Botong Wang,Singular Hodge theory for combinatorial geometries(2020). arXiv:2010.06088v4. [BL25] Spencer B...

work page arXiv 2026

[3] [3]

MR427110 [GL ¨O21] Cem G¨ uneri, San Ling, and Buket ¨Ozkaya,Quasi-cyclic codes, Concise encyclopedia of coding theory, 2021, pp. 129–149. MR4598739 [HH02] J¨ urgen Herzog and Takayuki Hibi,Discrete polymatroids, J. Algebraic Combin.16(2002), no. 3, 239–268 (2003). MR1957102 [HK12] June Huh and Eric Katz,Log-concavity of characteristic polynomials and the...

work page arXiv 2021

[4] [4]

(N.S.)27(2021), no

MR335518 [Las21] Micha l Laso´ n,On the toric ideals of matroids of a fixed rank, Selecta Math. (N.S.)27(2021), no. 2, Paper No. 18,

2021

[5] [5]

Math.259(2014), 1–12

MR4236584 [LM14] Micha l Laso´ n and Mateusz Micha lek,On the toric ideal of a matroid, Adv. Math.259(2014), 1–12. MR3197649 [Mas72] J. H. Mason,Matroids: unimodal conjectures and Motzkin’s theorem, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), 1972, pp. 207–220. MR0349445 [McG20] Sean McGuinness,Frame matroids, toric ideals,...

2014

[6] [6]

Simplex faces and quadratic toric ideals of lattice polytopes

MR4088420 [MO26] Aki Mori and Hidefumi Ohsugi,Simplex faces and quadratic toric ideals of lattice polytopes(2026). arXiv:2606.20430v1. [Rot71] Gian-Carlo Rota,Combinatorial theory, old and new, Actes du Congr` es International des Math´ ematiciens (Nice, 1970), Tome 3, 1971, pp. 229–233. MR505646 [Sch11] Jay Schweig,Toric ideals of lattice path matroids a...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[7] [7]

White,A unique exchange property for bases, Linear Algebra Appl.31(1980), 81–91

MR1363949 [Whi80] Neil L. White,A unique exchange property for bases, Linear Algebra Appl.31(1980), 81–91. MR570381 [YY25] Yu-Chuan Yu and Chi Ho Yuen,White’s conjecture for paving matroids(2025). arXiv:2510.04163v1

work page arXiv 1980