Given a finite regular graph

Abstract. We show that for every nondecreasing concave function ω: [0, ∞) → [0, ∞) with ω(0) = 0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X, ω◦d) for some metric d on X, or there exists α = α(ω)> 0 and n0 = n0(ω) ∈ N such that for all n � n0, any embedding of {0,..., n} ⊆ R into a metric space of the form (X, ω ◦ d) incurs distortion at least n α. 1.

We determine a class of rearrangement operators acting on dyadic intervals that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension to L p E, where E is a UMD space. We prove the existence of a large subspace Xp ⊂ Lp on which a bounded rearrangement operator acts as an isomorphism. Moreover, we study winning strategies for a combinatorial two person game played