Central Sets Theorem along filters and some combinatorial consequences

The Central Sets Theorem was introduced by H. Furstenberg and then afterwards several mathematicians have provided various versions extensions of this theorem. All these theorems deal with central sets, its origin from the algebra Stone–Čech compactification arbitrary semigroup, say βS. It can be proved that every closed subsemigroup βS is generated a filter. We will show that, under some restrictions, one derive for any theorem using corresponding filter algebra. Later we also how notions largeness along filters are preserved well behaved homomorphisms give consequences.