The graphs G(n, k) of the Johnson Schemes are unique for n ≥ 20

Let G(n, k) denote the graph of the Johnson Scheme J(n, k), i.e., the graph whose vertices are all k-subsets of a fixed n-set, with two vertices adjacent if and only if their intersection is of size k ? 1. It is known that G(n, k) is a distance regular graph with diameter k. Much work has been devoted to the question of whether a distance regular graph with the parameters of G(n, k) must isomorphic to G(n, k). In this paper, this question is settled affirmatively for n ≥ 20. In fact the result is proved with weaker conditions.