A measure-theoretic version of this theorem was used by H. Furstenberg in 1977 to give an elegant proof of Szemeredi's theorem: if you have a sequence of intege

A measure-theoretic version of this theorem was used by H. Furstenberg in 1977 to give an elegant proof of Szemeredi's theorem: if you have a sequence of integers of positive density, then it contains finite arithmetic progressions of all sizes. This is an example of Ramsey's principle [size => structure] of inevitable structure. The original proof by Szemeredi was combinatorial, and Furstenberg was among the first to use ergodic-theoretic methods to get theorems in arithmetic.