On Polynomial Extensions of van der Waerden's Theorem and its Applications

Abstract

In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem, where the recurrence set is guaranteed to be large in a precise combinatorial sense. As applications, we obtain new monochromatic polynomial configurations in both additive and multiplicative settings, including refined results over sum subsystems of IP-sets. Additionally, we prove exponential monochromatic patterns are abundant.