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Helly theorems for lattice of flats of a matroid
Abstract
A basis of a matroid is adapted to a flat if it contains a basis of the flat. We prove that a family of flats in a rank r matroid has a common adapted basis whenever every subfamily of at most r+1 flats does. The ground set and the family may be infinite; only the rank must be finite. This result, in particular, contains Klyachko's vector space Helly theorem for subspaces. As an application, we obtain a splitting criterion for tropical toric vector bundles on projective spaces. As a secondary application, we derive quadratic bounds for the Helly number for distributive generation in the lattice of flats.
Related
BibTeX
@misc{jafari2026helly,
title = {{Helly theorems for lattice of flats of a matroid}},
author = {Amir Jafari and Kiumars Kaveh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36566},
eprint = {2610.36566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36566},
}
RIS
TY - UNPB
TI - Helly theorems for lattice of flats of a matroid
AU - Amir Jafari
AU - Kiumars Kaveh
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36566
UR - https://arxiv.org/abs/2610.36566
AB - A basis of a matroid is adapted to a flat if it contains a basis of the flat. We prove that a family of flats in a rank r matroid has a common adapted basis whenever every subfamily of at most r+1 flats does. The ground set and the family may be infinite; only the rank must be finite. This result, in particular, contains Klyachko's vector space Helly theorem for subspaces. As an application, we obtain a splitting criterion for tropical toric vector bundles on projective spaces. As a secondary application, we derive quadratic bounds for the Helly number for distributive generation in the lattice of flats.
ER -