2022/11/09 by William J. Wesley, Wesley, William J. · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2211.05167
openalex publication_date 2022/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Van der Waerden's theorem states that for any positive integers k and r, there exists a smallest value n = w(k,r), called the van der Waerden number, such that every r-coloring of \1,…,n\ contains a monochromatic k-term arithmetic progression. We consider two variants of van der Waerden numbers: the numbers n = n(APD,k;r), the smallest value where every r-coloring of \1,…,n\ contains a monochromatic k-term arithmetic progression with common difference in D, and the numbers n = Δ(D,k;r), the smallest value n where every r-coloring of \1,…,n\ contains a sequence x1 < … < xk where the differences between consecutive terms are members of D. We study the case when D is set of Fibonacci numbers F and give improved bounds for the largest r where n(APF,k;r) and Δ(F,k;r) exist for all k. Moreover, we give some computational data on Δ(D,k;r) for other sets D.