The number of points in a matroid with no n-point line as a minor

TL;DR

This paper establishes an upper bound on the size of simple matroids avoiding a certain minor, showing that large such matroids are structurally similar to projective geometries over finite fields.

Contribution

It proves a bound on the number of points in matroids with no (l+2)-point line minor, characterizing extremal cases as projective geometries over GF(q).

Findings

Bound on matroid size in terms of prime powers.

Equality cases are projective geometries over GF(q).

Results extend understanding of matroid minors and structure.

Abstract

For any positive integer $l$ we prove that if $M$ is a simple matroid with no $(l+2)$-point line as a minor and with sufficiently large rank, then $|E(M)|\le \frac{q^{r(M)}-1}{q-1}$, where $q$ is the largest prime power less than or equal to $l$. Equality is attained by projective geometries over GF$(q)$.