9 research outputs found
New polynomial and multidimensional extensions of classical partition results
In the 1970s Deuber introduced the notion of -sets in
and showed that these sets are partition regular and contain all linear
partition regular configurations in . In this paper we obtain
enhancements and extensions of classical results on -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
. Second, we give two proofs of a generalization of Deuber's
results to general commutative semigroups. We also obtain a polynomial version
of the central sets theorem of Furstenberg, extend the theory of
-systems of Deuber, Hindman and Lefmann and generalize a classical
theorem of Rado regarding partition regularity of linear systems of equations
over to commutative semigroups.Comment: Some typos, including a terminology confusion involving the words
`clique' and `shape', were fixe
Dynamical characterization of C-sets and its application
In this paper, we set up a general correspondence between the algebra
properties of \bN and the sets defined by dynamical properties. In
particular, we obtain a dynamical characterization of C-sets, where C-sets are
the sets satisfying the strong Central Sets Theorem. As an application, we show
that Rado systems are solvable in C-sets.Comment: 30 pages, mirror changes, to appear in Fund. Mat