New polynomial and multidimensional extensions of classical partition results

摘要

In the 1970s Deuber introduced the notion of (m, p, c)-sets in N and showed that these sets are partition regular and contain all linear partition regular configurations in N. In this paper we obtain enhancements and extensions of classical results on (m,p, c) -sets in two directions. First, we show, with the help of ultrafilter techniques, that Deuber's results extend to polynomial configurations in abelian groups. In particular, we obtain new partition regular polynomial configurations in Z(d). Second, we give two proofs of a generalization of Deuber's results to general commutative semigroups.