Abstract: Consider a measure-preserving action of a countable group G on a standard probability space and suppose that this action is mixing along a given filter F on G; e.g. mild mixing corresponds to F=IP*, while letting F be the filter of sets of upper density 1 gives weak mixing for amenable G. These notions of mixing imply double recurrence for such systems, and it is a major theme in ergodic Ramsey theory to amplify this to multiple recurrence. This is often done using a so-called van der Corput difference (ratio) lemma, which "drops" the degree of the recurrence, thus enabling proofs by induction. Analogues of this lemma had been proven for various filters, but the existing proofs were different in each case. We prove a van der Corput lemma for a general class of filters, which includes those mentioned above, as well as idempotent ultrafilters. This is based on a new Ramsey theorem for semigroups related to labeling edges between the semigroup elements with their ratios. --- This is a continuation of the talk given in the Logic seminar on Tuesday February 11th.