Graphs and Matroids Seminar - Jane Gao

Title: Minors of random representable matroid over finite fields

Abstract: Consider a random n by m matrix A over GF(q) where every column has k nonzero elements, and let M[A] be the matroid represented by A. In the case that q=2, Cooper, Frieze and Pegden (RSA 2019) proved that given a fixed binary matroid N, if k is sufficiently large, and m/n is sufficiently large (both depending on N), then whp. M[A] contains N as a minor. We improve their result by determining the sharp threshold (of m/n) for the appearance of a fixed q-nary matroid N as a minor of M[A], for every k\ge 3, and every prime q. This is joint work with Peter Nelson.