Rearrangements of incidence tables

AbstractGiven integers 0 < k ⩽ n and a permutation α mapping the set of integers from 1 to n onto itself, there exists a permutation β mapping the set of unordered pairs of integers from 1 to n onto itself in such a way that whenever 0 < |a − b| ⩽ k and {aα, bα}β = {x, y}, then also 0 < |x − y| ⩽ k and x and y are equal to or between aα and bα