Avoiding Monochromatic Sequences With Special Gaps

2003/02/04 by Bruce M. Landman, Bruce Landman, Aaron Robertson +2 · 1 citation
Engineering · Mathematics · #05D10 #11B25 #11N13 #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems #math.CO #msc:05D10 #msc:11B25 #msc:11N13

paper · pdf · doi:10.48550/arxiv.math/0302041

16 pages

arxiv created 2003/02/04 · openalex publication_date 2003/02/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For S a set of positive integers, and k and r fixed positive integers, denote by f(S,k;r) the least positive integer n (if it exists) such that within every r-coloring of \1,2,...,n\ there must be a monochromatic sequence \x₁,x₂,...,x_k\ with x_i-x_i-1 ∈ S for 2 ≤ i ≤ k. We consider the existence of f(S,k;r) for various choices of S, as well as upper and lower bounds on this function. In particular, we show that this function exists for all k if S is an odd translate of the set of primes and r=2.

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