2012/06/27 by Nikhil Bansal, Bansal, N., Rudi Pendavingh +3 · 2 citations
Computer Science · #05B35 matroids and geometric lattices #05D40 probabilistic methods #52C45 combinatorial complexity of geometrical structures #Advanced Graph Theory Research #Combinatorics (math.CO) #Complexity and Algorithms in Graphs #Computational Geometry and Mesh Generation #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1206.6270
openalex publication_date 2012/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of determining mn, the number of matroids on n elements. The best known lower bound on mn is due to Knuth (1974) who showed that log log mn is at least n-3/2log n-1. On the other hand, Piff (1973) showed that loglog mn≤ n-log n+loglog n +O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on loglog mn that is within an additive 1+o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.