## A new and stronger central sets theorem

### Volume 199 / 2008

#### Abstract

Furstenberg's original *Central Sets Theorem* applied
to *central* subsets of $\mathbb N$ and finitely many specified
sequences in $\mathbb Z$. In this form it was already strong enough to
derive some very strong combinatorial consequences, such as the
fact that a central subset of $\mathbb N$ contains solutions to
all partition regular systems of homogeneous equations. Subsequently the
Central Sets Theorem was extended to apply to arbitrary semigroups and
countably many specified sequences. In this paper we derive
a new version of the Central Sets Theorem
for arbitrary semigroups $S$ which applies to *all*
sequences in $S$ at once. We show that the new version is strictly stronger
than the original version applied to the semigroup $(\mathbb R,+)$. And we show that
the noncommutative versions are strictly increasing in strength.