Cofinalities of elementary substructures of structures on א_ω

We give some sufficient conditions for {a ⊆ א_ω: cf(sup(a∩א_n)) = f(n) for n < ω} to be stationary. Thus we get consistency e.g. of cases where f has ≥ three values attained infinitely often. We use pcf, partition theorems on trees and games.