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

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$(q)$

On minor-closed classes of matroids with exponential growth rate

Let \cM be a minor-closed class of matroids that does not contain arbitrarily long lines. The growth rate function, h:\bN\rightarrow \bN of \cM is given by h(n) = \max(|M|\, : \, M\in \cM, simple, rank-$n$). The Growth Rate Theorem shows that there is an integer $c$ such that either: $h(n)\le c\, n$, or ${n+1 \choose 2} \le h(n)\le c\, n^2$, or there is a prime-power $q$ such that $\frac{q^n-1}{q-1} \le h(n) \le c\, q^n$; this separates classes into those of linear density, quadratic density, and base-$q$ exponential density. For classes of base-$q$ exponential density that contain no $(q^2+1)$-point line, we prove that $h(n) =\frac{q^n-1}{q-1}$ for all sufficiently large $n$. We also prove that, for classes of base-$q$ exponential density that contain no $(q^2+q+1)$-point line, there exists k\in\bN such that $h(n) = \frac{q^{n+k}-1}{q-1} - q\frac{q^{2k}-1}{q^2-1}$ for all sufficiently large $n$

Counting matroids in minor-closed classes

A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without $M(K_4)$-minor which asymptoticaly matches the best known lower bound on the number of all matroids, due to Knuth.Comment: 13 pages, 3 figure

Growth rate functions of dense classes of representable matroids

AbstractFor each proper minor-closed subclass M of the GF(q2)-representable matroids containing all GF(q)-representable matroids, we give, for all large r, a tight upper bound on the number of points in a rank-r matroid in M, and give a rank-r matroid in M for which equality holds. As a consequence, we give a tight upper bound on the number of points in a GF(q2)-representable, rank-r matroid of large rank with no PG(k,q2)-minor