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KKM theorems and discrete geometry beyond matroids
Abstract
We introduce selection structures, a topological framework that extends the role played by color classes and matroids in discrete geometry and KKM theorems. Selection structures allow us to extend classic results to genuinely non-matroidal examples, including chessboard complexes and matching complexes. We show that several matroidal versions of classic results can be generalized to selection structures. These include McGinnis' version of Komiya's KKMS theorem, Holmsen's version of Carathéodory's theorem, Kalai and Meshulam's version of Helly's theorem, and Sadovek's version of the Goodman--Pollack transversal theorem.
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BibTeX
@misc{soberon2026theorems,
title = {{KKM theorems and discrete geometry beyond matroids}},
author = {Pablo Soberón},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.16832},
eprint = {2606.16832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.16832},
}
RIS
TY - UNPB
TI - KKM theorems and discrete geometry beyond matroids
AU - Pablo Soberón
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2606.16832
UR - https://arxiv.org/abs/2606.16832
AB - We introduce selection structures, a topological framework that extends the role played by color classes and matroids in discrete geometry and KKM theorems. Selection structures allow us to extend classic results to genuinely non-matroidal examples, including chessboard complexes and matching complexes. We show that several matroidal versions of classic results can be generalized to selection structures. These include McGinnis' version of Komiya's KKMS theorem, Holmsen's version of Carathéodory's theorem, Kalai and Meshulam's version of Helly's theorem, and Sadovek's version of the Goodman--Pollack transversal theorem.
ER -