Gr\"obner bases of toric ideals associated with matroids

In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gr"obner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gr"obner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.