The sharp threshold for the multiplicative Merino-Welsh inequality

Merino and Welsh conjectured that a loopless bridgeless graph satisfies $\max(T_G(2,0), T_G(0,2)) \ge T_G(1,1)$ . Jackson proved the multiplicative matroid form $T_M(3,0)T_M(0,3) \ge T_M(1,1)^2$ , and the constant 3 was lowered to $2.9243$ and then to $2.355$ by Csikvári. Counterexamples are known below $x_* = 2.2266815969\ldots$ , the largest real root of $x^3 = 9(x-1)$ , and Csikvári conjectured in 2025 that $x_*$ is exactly the threshold: does $T_M(x_*,0)T_M(0,x_*) \ge T_M(1,1)^2$ hold for every finite matroid without loops or coloops?

Result: Proved(see note)
Status: Resolved
AI contribution: AI co-developed
Method: Argument
Field: Matroid theory; Tutte polynomials
Posed by: Péter Csikvári, Conjecture 7.1 of Around the Merino-Welsh conjecture: improving Jackson's inequality (arXiv:2502.19196, 2025); the underlying conjecture is Merino and Welsh's
Year posed: 2025
Years open: 1y
Solved: 2026-09
Model: GPT-6 Astra; Claude Opus 5
Vendor: OpenAI; Anthropic
Collaborators: Mingchang Liu
Verification: Unreviewed
Publication: Preprint
Significance: 14 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

It is proved that $T_M(x_*,0)T_M(0,x_*)\ge T_M(1,1)^2$ for every finite matroid $M$ without loops or coloops, and known counterexamples below $x_*$ make the universal threshold sharp. For $2\le x\le x_*$ , the stronger bound $\frac{T_M(x,0)T_M(0,x)}{T_M(1,1)^2}\ge\left(\frac{x^3}{9(x-1)}\right)^{(m-2\kappa)/2}$ holds, where $m$ is the number of elements and $\kappa$ the number of connected components. Parallel doublings of uniform matroids show the exponential rate is optimal throughout the interval. The proof uses weighted broken circuits, matroid decomposition, and exact checks of rational replacements. The graph conjecture at $x=2$ remains open.

What the AI did

This paper grew out of a human–AI collaboration on the Merino–Welsh problem. The author chose the sharp universal matroid threshold and its quantitative refinement as the paper’s focus, directed the comparison with earlier work, and shaped the exposition through successive revisions. AI models contributed to mathematical exploration, proof development, and exact computation, including the rational replacement table. In preparing the manuscript, GPT-6 Astra (OpenAI) and Claude Opus 5 (Anthropic) assisted with checking the arguments, researching the literature, and producing the text, code, and figures.

Verification

Checked here on 27 September 2026. Csikvári's Conjecture 7.1 was located in arXiv:2502.19196v2, the version the submission links, and the Zenodo record 22911905 of 23 September was read: the threshold is stated as $x_* = 2.2266815969\ldots$ , the largest real root of $x^3 = 9(x-1)$ , with the quantitative bound on $2 \le x \le x_*$ and sharpness of the exponential rate from parallel doublings of uniform matroids. The tools named are a weighted broken-circuit comparison, the Cunningham-Edmonds decomposition and exact polynomial checks of a finite rational replacement table; the table and a checker are in the linked V1.1 release. The mathematics was not checked here, the rational table was not recomputed, and the author's own note is correct that critical AI reviews are not verification. Four days old, no referee.

Sources

Submitted by SilentIbis759 on 23 Sep 2026

The sharp threshold for the multiplicative Merino-Welsh inequality

What was actually shown

What the AI did

Verification

Sources

Discussion