Isolating subgaps of a multiple gap

Abstract

Given an analytic multiple gap Γ=Γi:i<n, we study the family B(Γ) of the sets A⊂ n for which there is a restriction Γi|a:i∈A which is still a multiple gap, while a∈Γi⊥ for i∉ A. This family always contains at least two sets of cardinality 2, and every set of cardinality k is contained in a set from B(Γ) of cardinality J(k), a number that grows as 382πk·9k. All these results can be stated in terms of the topology of the Čech–Stone remainder ω∗ and in terms of sequences in Banach spaces. For example, for any finite family of analytic open sets of ω∗ with non-disjoint closures there is always a point that lies in exactly two closures. And given a sequence xn n<ω of vectors in a Banach space that contains subsequences equivalent to ℓ1, ℓ2, … , ℓn in a way that cannot be separated, it always contains a subsequence xnk k<ω where the ℓ1 and ℓ2 subsequences cannot be separated, while there are at most 6 (and this is sharp) of the remaining p’s for which xnk k<ω contains subsequences equivalent to ℓp.