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 \mathcal {M} is the function h M ( n ) h_{\mathcal {M}}(n) whose value at an integer n ≥ 0 n \ge 0 is defined to be the maximum number of elements in a simple matroid in M \mathcal {M} of rank at most n n . Geelen, Kabell, Kung and Whittle showed that, whenever h M ( 2 ) h_{\mathcal {M}}(2) is finite, the function h M h_{\mathcal {M}} grows linearly, quadratically or exponentially in n n (with base equal to a prime power q q ), up to a constant factor. We prove that in the exponential case, there are nonnegative integers k k and d ≤ q 2 k − 1 q − 1 d \le \tfrac {q^{2k}-1} {q-1} such that h M ( n ) = q n + k − 1 q − 1 − q d h_{\mathcal {M}}(n) = \frac {q^{n+k}-1}{q-1} - qd for all sufficiently large n n , and we characterise which matroids attain the growth rate function for large n n . We also show that if M \mathcal {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 k and d d , as well as the point that ‘sufficiently large’ begins to apply to n n , can be determined by a finite computation.