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arXiv · 2609.38047

Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats

Kolja Knauer·Leonardo Martínez-Sandoval·Criel Merino
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Abstract

Beke, Csáji, Csikvári, and Pituk showed that the Merino--Welsh quotient $Φ(M)=T_M(1,1)^2/(T_M(2,0)T_M(0,2))$ can be arbitrarily large, so the multiplicative Merino-Welsh inequality fails for matroids in general. We show that $Φ$ is uniformly bounded on the class of matroids whose cyclic-flat lattice avoids any fixed finite poset $P$ as an induced subposet. We further prove $Φ(M)\leq1$ for matroids of cyclic width at most $7$, cyclic height at most $6$, and for loop- and coloop-free $4$-paving or $4$-copaving matroids.

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Kolja Knauer, Leonardo Martínez-Sandoval, Criel Merino. 2026-09-29. Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats. https://arxiv.org/abs/2609.38047

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