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Unavoidable flats in GF(q)-representable matroids
Abstract
We show that, for any finite field π½ and positive integer k, a simple π½-representable matroid M with sufficiently high rank contains a rank-k flat which is either independent in M, or is a projective or affine geometry over a subfield of π½. This generalizes a result for matroids representable over prime fields due to the first two authors. As a corollary, we obtain a Ramsey theorem for matroids representable over a finite field π½. For an integer k, in any $2$-colouring of the points of an π½-representable matroid of sufficiently high rank, there is a monochromatic rank-k flat.
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BibTeX
@misc{geelen2026unavoidable,
title = {{Unavoidable flats in GF(q)-representable matroids}},
author = {Jim Geelen and Matthew E. Kroeker and Peter Nelson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22135},
eprint = {2610.22135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22135},
}
RIS
TY - UNPB
TI - Unavoidable flats in GF(q)-representable matroids
AU - Jim Geelen
AU - Matthew E. Kroeker
AU - Peter Nelson
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22135
UR - https://arxiv.org/abs/2610.22135
AB - We show that, for any finite field π½ and positive integer k, a simple π½-representable matroid M with sufficiently high rank contains a rank-k flat which is either independent in M, or is a projective or affine geometry over a subfield of π½. This generalizes a result for matroids representable over prime fields due to the first two authors. As a corollary, we obtain a Ramsey theorem for matroids representable over a finite field π½. For an integer k, in any $2$-colouring of the points of an π½-representable matroid of sufficiently high rank, there is a monochromatic rank-k flat.
ER -