On generalizing the Van der Waerden theorem to some symmetric functions

Abstract

Let $n,m$ be positive integers and $c \in \mathbb{Z}_n$, where $\mathbb{Z}_n$ is the ring of integers modulo $n$. We almost complete providing the answer to the following problem, partially solved by N. Alon. Does any infinite sequence over $\mathbb{Z}_n$ contain $m$ same-length consecutive blocks $B_1, \ldots, B_m$ s.t. $\sum B_j + c \prod B_j = 0$ for every $j=1,\ldots,m$ (where $\sum B$ and $\prod B$ denote, respectively, the sum and the product of the elements in block $B$)? In the case of $c=0$, this problem is equivalent to the Van der Waerden theorem. After investigating $B \mapsto \sum B + c\prod B$, we provide other examples of generalizing the Van der Waerden theorem.