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A combinatorial approach to central to theorem

H. Furstenberg introduced the notion of central set in terms of topological dynamics and established the central set theorem. The essence of central set theorem is that it is the simultaneous extension of van der Waerden's theorem and Hindman's theorem. Later V. Bergelson and N. Hindman established a connection between central sets and the algebra of Stone-cech compactification and proved that the central sets are the member of minimal idempotent ultrafilter of discrete semigroup. In some subsequent papers the central set theorem has been studied more deeply and some generalizations has been established. Those works use the techniques of Stone-Cech compactification of discrete semigroup. In this work we will prove central set theorem via combinatorial approach using the combinatorial characterization of central set established by N. Hindman, A. Malkeki and D. Strauss. Though we will use the combinatorial approach but the technique of the proof is similar to the proof of D. De, N. Hindman, D. Strauss.