On generalizing the Van der Waerden theorem to some symmetric functions

2025/02/11 by Arie Bialostocki, Bialostocki, Arie, Vladyslav Oles +1
Computer Science · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2502.07921

openalex publication_date 2025/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let n,m be positive integers and c ∈ ℤ_n, where 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 ℤ_n contain m same-length consecutive blocks B₁, …, B_m s.t. ∑ B_j + c ∏ B_j = 0 for every j=1,…,m (where ∑ B and ∏ 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 ↦ ∑ B + c∏ B, we provide other examples of generalizing the Van der Waerden theorem.

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