Automorphisms of infinite Johnson graphs

Let I be a set of infinite cardinality α . For every cardinality β ≤ α the Johnson graphs J β and J β are the graphs whose vertices are subsets X ⊂ I satisfying | X | = β , | I ∖ X | = α and | X | = α , | I ∖ X | = β (respectively) and vertices X , Y are adjacent if | X ∖ Y | = | Y ∖ X | = 1 . Note that J α = J α and J β is isomorphic to J β for every β < α . If β is finite then J β and J β are connected and it is not difficult to prove that their automorphisms are induced by permutations on I . In the case when β is infinite, these graphs are not connected and we determine the restrictions of their automorphisms to connected components.