2014/09/02 by Geelen, Jim, Nelson, Peter · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1409.0777
The growth-rate function for a minor-closed class M of matroids is the function h where, for each non-negative integer r, h(r) is the maximum number of elements of a simple matroid in M with rank at most r. The Growth-Rate Theorem of Geelen, Kabell, Kung, and Whittle shows, essentially, that the growth-rate function is always either linear, quadratic, exponential, or infinite. Morover, if the growth-rate function is quadratic, then h(r)≥ \binomr+12, with the lower bound coming from the fact that such classes necessarily contain all graphic matroids. We characterise the classes that satisfy h(r) = \binomr+12 for all sufficiently large r.