VIP systems in partial semigroups
Abstract A VIP system is a polynomial-type generalization of the notion of an IP system, i.e., a set of finite sums. We extend the notion of VIP system to commutative partial semigroups and obtain an analogue of Furstenberg's central sets theorem for these systems which extends the polynomial Hales–Jewett Theorem of Bergelson and Leibman. Several Ramsey theoretic consequences, including the central sets theorem itself, are then derived from these results.