Partition Regular Structures Contained in Large Sets Are Abundant

Furstenberg and Glasner have shown that for a particular notion of largeness in a group, namely piecewise syndeticity, if a set B is a large subset Z, then for any l∈N, the set of length l arithmetic progressions lying entirely in B is large among the set of all length l aritmetic progressions. We extend this result to apply to infinitely many notions of largeness in arbitrary semigroups and to partition regular structures other than arithmetic progressions. We obtain, for example, similar results for the Hales–Jewett theorem.