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Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats
Abstract
Beke, Csáji, Csikvári, and Pituk showed that the Merino--Welsh quotient Φ(M)=TM(1,1)2/(TM(2,0)TM(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)≤1 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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BibTeX
@misc{knauer2026merinowelsh,
title = {{Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats}},
author = {Kolja Knauer and Leonardo Martínez-Sandoval and Criel Merino},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38047},
eprint = {2609.38047},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38047},
}
RIS
TY - UNPB
TI - Merino-Welsh inequalities for matroids with controlled lattices of cyclic flats
AU - Kolja Knauer
AU - Leonardo Martínez-Sandoval
AU - Criel Merino
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38047
UR - https://arxiv.org/abs/2609.38047
AB - 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)≤1 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.
ER -