A new and stronger central sets theorem
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.