On generalizing the Van der Waerden theorem to some symmetric functions

TL;DR

This paper extends the Van der Waerden theorem to a class of symmetric functions involving sums and products over blocks in infinite sequences over modular integers, partially solving a problem posed by N. Alon.

Contribution

It introduces new generalizations of the Van der Waerden theorem involving functions combining sums and products of sequence blocks.

Findings

Almost complete solution to the problem for certain functions

Identification of new classes of functions generalizing Van der Waerden

Examples of functions beyond the sum that satisfy similar properties

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.