On generalizing the Van der Waerden theorem to some symmetric functions
Arie Bialostocki, Vladyslav Oles
TL;DR
This work studies a zero-sum variant of the Van der Waerden theorem for blocks in sequences over $\mathbb{Z}_n$ using the map $B \mapsto \sum B + c \prod B$. It provides a complete classification for $c=1$ across small moduli and sharp non-vanishing results for many other $n$, while also addressing $c=-1$ with a prime-power–level classification and composite-case obstructions; it then extends the framework to multivariate, vector-valued transformation sums and connects these to power sums and elementary symmetric polynomials via Newton identities. The methods combine Alon-style zero-sum colorings, Thue-type constructions, and modular factorization arguments to illuminate how the arithmetic of $n$ controls vanishing behavior. Together, these results generalize Van der Waerden-type zero-sum phenomena to symmetric-polynomial transformations and to multivariate settings, enriching Ramsey theory on words with modular-algebraic structure.
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.