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

• Published in: Transactions of the American Mathematical Society 2017-09, Vol.369 (9), p.6751-6776
• Main Authors: Geelen, Jim, Nelson, Peter
• Format: Article
• Language: English
• Subjects: Research article
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7186

The growth rate function for a nonempty minor-closed class of matroids M\mathcal {M} is the function hM(n)h_{\mathcal {M}}(n) whose value at an integer n≥0n \ge 0 is defined to be the maximum number of elements in a simple matroid in M\mathcal {M} of rank at most nn. Geelen, Kabell, Kung and Whittle showed that, whenever hM(2)h_{\mathcal {M}}(2) is finite, the function hMh_{\mathcal {M}} grows linearly, quadratically or exponentially in nn (with base equal to a prime power qq), up to a constant factor. We prove that in the exponential case, there are nonnegative integers kk and d≤q2k−1q−1d \le \tfrac {q^{2k}-1} {q-1} such that hM(n)=qn+k−1q−1−qdh_{\mathcal {M}}(n) = \frac {q^{n+k}-1}{q-1} - qd for all sufficiently large nn, and we characterise which matroids attain the growth rate function for large nn. 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 kk and dd, as well as the point that ‘sufficiently large’ begins to apply to nn, can be determined by a finite computation.