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Recently Di Nasso and Ragosta proved for every finite partition of$\mathbb{N}$, there exists a monochromatic infinite exponential pattern of asingle sequence. In this article, we prove these patterns are abundant. We showeach term of this pattern can be chosen from a finite set, consists of amultiplicative $IP_r$ set and a geometric progression of finite length. Toprove this, first we improve the size of the recurrence set of the polynomialVan der Waerden's Theorem, and then applying the size of the recurrence set, weprove the existence of some new monochromatic polynomial patterns. Then,applying methods of induction, we construct such infinitary patterns.Throughout our article, we use the methods from the Algebra of theStone-\v{C}ech-Compactification of the set of natural numbers.

Large recurrence of the polynomial Van der Waerden's theorem and its application to monochromatic exponential patterns