Partition regular structures contained in large sets are abundant

Furstenberg and Glasner have shown that for a particular notion of largenes s in a group, namely piecewise syndetiety, if a set B is a large subset Z, then for any l is an element of N, the set of length l arithmetic progressi ons lying entirely in B is large among the set of all length l aritmetic pr ogressions. We extend this result to apply to infinitely many notions of la rgeness in arbitrary semigroups and to partition regular structures other t han arithmetic progressions. We obtain, for example, similar results for th e Hales Jewett theorem. (C) 2001 Academic Press.