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Highly Connected Counterexamples to the Multiplicative Merino–Welsh Inequality

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Jungang Chen ↗

Version 1 / Oct 21, 2026 / CC BY 4.0

Abstract

For every integer k≥3k\ge 3, we construct kk-connected matroids representable over every infinite field, with girth kk, arbitrarily large cogirth, and Merino–Welsh quotient \[ \Phi(M)=\frac{T_M(1,1)^2}{T_M(2,0)T_M(0,2)} \] tending to infinity. In particular, this disproves **Conjecture 7.2 of Csikvári (2026)**, which asserts the multiplicative Merino–Welsh inequality for simple and cosimple matroids, and therefore also the stronger **Conjecture 7.3** therein, as observed by its author. Our examples are rank truncations of direct sums of kk-element circuits. We give a 3-connected counterexample on 1011 elements of rank 433, certified by exact integer arithmetic, and a 6048-element counterexample admitting a proof without computation. Taking the direct sum of a counterexample with its dual also gives a negative answer to **Problem 4.3 of Beke, Csáji, Csikvári, and Pituk (2024)**: the original maximum and additive Merino–Welsh inequalities can fail for simple and cosimple matroids. The same construction disproves **Conjecture 2 of Knauer, Martínez-Sandoval, and Merino (2026)**: for every fixed g≥3g\ge 3, Φ\Phi is unbounded even among matroids having both girth and cogirth exactly gg. The dual direct-sum examples need not be connected; these constructions do not disprove the graphical or binary versions.

Provenance statement

This work was initiated by the human contributor, who posed the research problem and directed the investigation. OpenAI ChatGPT (GPT-6) played a substantial role in developing the counterexample constructions, deriving the mathematical arguments, identifying related conjectures, performing literature searches, and preparing the manuscript and computational verification code. The manuscript is primarily AI-generated. The explicit finite counterexample has been checked using exact integer arithmetic, and the mathematical arguments have undergone AI-assisted consistency checks. However, these checks do not constitute independent human verification or formal proof certification. The complete proofs have not yet been independently verified by a human. This work is made publicly available to facilitate mathematical scrutiny and further verification.

Tools used

OpenAI
ChatGPTVersion 6 Pro

References

  1. C. Beke, G. K. Csáji, P. Csikvári, and S. Pituk, The Merino--Welsh conjecture is false for matroids , Advances in Mathematics 446 (2024), 109674. doi:10.1016/j.aim.2024.109674 (https://doi.org/10.1016/j.aim.2024.109674). The published version contains Problem 4.3.DOI
  2. P. Csikvári, Around the Merino--Welsh conjecture: improving Jackson's inequality , European Journal of Combinatorics 137 (2026), 104402. doi:10.1016/j.ejc.2026.104402 (https://doi.org/10.1016/j.ejc.2026.104402). Conjecture 7.2 is stated in arXiv:2502.19196v2 (https://arxiv.org/abs/2502.19196).arXivDOI
  3. K. Knauer, L. Martínez-Sandoval, and C. Merino, Merino--Welsh inequalities for matroids with controlled lattices of cyclic flats , arXiv:2609.38047 (https://arxiv.org/abs/2609.38047), version 1, 29 September 2026. Conjecture 2.arXiv

Version history

  1. v1Submitted by Jungang ChenInitial depositCurrentOct 21, 2026