The Smallest Matroids with no Large Independent Flat

Peter Nelson,Sergey Norin

doi:10.37236/8992

Abstract

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.

Highlights

Call a set S in a matroid M a claw of M if S is both a flat and an independent set of M
These objects were introduced by Bonamy et al [2] and studied by Nelson and Nomoto [3]; both of these papers consider the structure of 3-claw-free binary matroids
We deal with general matroids, and address the simple extremal question of determining the smallest simple rank-r matroids omitting a given claw; we solve this problem and characterize the tight examples

Summary

Introduction

Call a set S in a matroid M a claw of M if S is both a flat and an independent set of M. In the context of these works, the graph-theoretic notions of induced subgraphs, cliques, chromatic number and forests have analogies in the setting of simple binary matroids: cliques are analogous to projective geometries in the sense of being maximal with a given rank, while claws correspond to induced forests. A graph-theoretic analogue of Theorem 1 using this correspondence would characterize graphs on r vertices with minimum number of edges and no induced forests of given size. If M is a simple triangle-free matroid with no (2t + 1)-claw, |M | t2r/t−1 This conjectured bound holds with equality when M is the direct sum of t copies of a rank-(r/t) binary affine geometry; these should be the only cases where equality holds.

The Bound

Equality

Graphs