We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined to be the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$. We also show for $r \geqslant 2t$ that $M_{r,t}$ is the unique example for which equality holds.
@article{10_37236_8992,
author = {Peter Nelson and Sergey Norin},
title = {The smallest matroids with no large independent flat},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {1},
doi = {10.37236/8992},
zbl = {1458.05039},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8992/}
}
TY - JOUR
AU - Peter Nelson
AU - Sergey Norin
TI - The smallest matroids with no large independent flat
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8992/
DO - 10.37236/8992
ID - 10_37236_8992
ER -
%0 Journal Article
%A Peter Nelson
%A Sergey Norin
%T The smallest matroids with no large independent flat
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8992/
%R 10.37236/8992
%F 10_37236_8992
Peter Nelson; Sergey Norin. The smallest matroids with no large independent flat. The electronic journal of combinatorics, Tome 28 (2021) no. 1. doi: 10.37236/8992
