A direct inductive proof of the sharp Merino--Welsh threshold for matroids
Abstract
Motivated by the Merino--Welsh conjecture, we consider the smallest c≥0, denoted by c*, for which the inequality TM(c,0)TM(0,c)≥ TM(1,1)2 holds for every loopless and coloopless finite matroid M. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [\emph{Adv. Math.} \textbf{446} (2024), 109674] give the lower bound x0, where x0≈2.22668 is the largest real root of the polynomial x3−9(x−1). Later, Csikvári [\emph{European J. Combin.} \textbf{137} (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at c=x0. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that c*=x0, without computer-assisted finite verification.
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BibTeX
@misc{chen2026direct,
title = {{A direct inductive proof of the sharp Merino--Welsh threshold for matroids}},
author = {Jungang Chen and Jiaxin Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06589},
eprint = {2610.06589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06589},
}
RIS
TY - UNPB
TI - A direct inductive proof of the sharp Merino--Welsh threshold for matroids
AU - Jungang Chen
AU - Jiaxin Xie
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.06589
UR - https://arxiv.org/abs/2610.06589
AB - Motivated by the Merino--Welsh conjecture, we consider the smallest c≥0, denoted by c_*, for which the inequality T_M(c,0)T_M(0,c)≥ T_M(1,1)^2 holds for every loopless and coloopless finite matroid M. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [\emph{Adv. Math.} \textbf{446} (2024), 109674] give the lower bound x_0, where x_0≈2.22668 is the largest real root of the polynomial x^3−9(x−1). Later, Csikvári [\emph{European J. Combin.} \textbf{137} (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at c=x_0. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that c_*=x_0, without computer-assisted finite verification.
ER -