ISOPERIMETRIC-INEQUALITIES AND FRACTIONAL SET SYSTEMS

Let Ω be the probability space of all 0-1 sequences of length n, with P((ai)1n) = pΣa(1 - p)n - Σa. For a set A ⊂ Ω and a natural number t, let A(t) be the set of sequences with Hamming distance at most t from A. The main aim of this paper is to prove that if A is a down-set and P(A)≥σkr=0 ( n r) pr (1 - p) n-r then P(A(t))≥σk+1r=0 ( n r) pr (1 - p)n-r. This result generalises Harper's theorem on the isoperimetric inequality in the cube. © 1991.