On the toric ideals of matroids of a fixed rank

In $1980$ White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White's conjecture for high degrees with respect to the rank. This extends our result arXiv:1302.5236 confirming White's conjecture `up to saturation'. Furthermore, we study degrees of Gr\"{o}bner bases and Betti tables of the toric ideals of matroids of a fixed rank.
Comment: final version, 16 pages