Partition theorems for layered partial semigroups

Abstract

We introduce the notions of layered semigroups and partial semigroups, and prove some Ramsey type partition results about them. These results generalize previous results of Gowers, Furstenberg, and of Bergelson, Blass, and Hindman. We give some applications of these results (see, e.g., Theorem 1.1) and present examples suggesting that our results are rather optimal.