Gowers' Ramsey Theorem for generalized tetris operations

Abstract

We prove a generalization of Gowers' theorem for FIN_k where, instead of the single tetris operation T:FIN_k→FIN_(k−1), one considers all maps from FIN_k to FIN_j for 0≤j≤k arising from nondecreasing surjections f:{0,1,…,k}→{0,1,…,j}. This answers a question of Bartošová and Kwiatkowska. We also describe how to prove a common generalization of such a result and the Galvin–Glazer–Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.

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