Ramsey transfer to semi-retractions

Abstract

We introduce the notion of a semi-retraction. Given two structures A and B, A is a semi-retraction of B if there exist quantifier-free type respecting maps f:B→A and g:A→B such that f∘g is an embedding. We say that a structure has the Ramsey property if its age does. Given two locally finite ordered structures A and B, if A is a semi-retraction of B and B has the Ramsey property, then A also has the Ramsey property. We introduce notation for what we call semi-direct product structures, after the group construction known to preserve the Ramsey property [11]. We introduce the notion of a color-homogenizing map, and use this notion to give a finitary argument that the semi-direct product structure of ordered relational structures with the Ramsey property must also have the Ramsey property. Finally, we characterize NIP theories using a generalized indiscernible sequence indexed by a semi-direct product structure.