The densest matroids in minor-closed classes with exponential growth rate

Abstract

The growth rate function for a nonempty minor-closed class of matroids M is the function h M (n) whose value at an integer n ≥ 0 is defined to be the maximum number of elements in a simple matroid in M of rank at most n. Geelen, Kabell, Kung and Whittle showed that, whenever h M (2) is finite, the function h M grows linearly, quadratically or exponentially in n (with base equal to a prime power q), up to a constant factor.We prove that in the exponential case, there are nonnegative integers k and d ≤ q 2k −1 q−1 such that h M (n) = q n+k −1 q−1 − qd for all sufficiently large n, and we characterise which matroids attain the growth rate function for large n. We also show that if M is specified in a certain 'natural' way (by intersections of classes of matroids representable over different finite fields and/or by excluding a finite set of minors), then the constants k and d, as well as the point that 'sufficiently large' begins to apply to n, can be determined by a finite computation.

Excluding a line from $\mathbb C$-representable matroids

“…In other words, there is ample room to strengthen and refine the statement. For instance, [14] does this for classes satisfying (3), showing that there are in fact integers k, d ≥ 0 such that h M (r) = q r+k −1 q−1 − qd, for all large r. We focus on classes satisfying (2), which are called quadratically dense classes. Our main result, Theorem 1.0.9, substantially refines outcome (2), and has Theorems 1.0.5, 1.0.6 and 1.0.7 as corollaries.…”

“…where the first inequality holds because r A +r B −s ≤ r N and p(x−s) ≤ p(x−s+1) for all x ≥ r1 , and the last inequality holds by 6. Stacks were used to find the extremal functions for exponentially dense minor-closed classes in [12] and [14] with O as the class of GF(q)representable matroids; our definition generalizes the original definition from [11]. We say that a matroid M is an O-stack if there are integers b ≥ 2 and h ≥ 1 so that M is an (O, b, h)-stack.…”