Automorphisms of infinite Johnson graphs

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