Unrolling residues to avoid progressions

2012/09/12 by Steve Butler, Butler, Steve, Ron Graham +3
Computer Science · Engineering · Mathematics · #05D10 #11A15 #11B25 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1209.2687

openalex publication_date 2012/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of coloring [n]=1,2,...,n with r colors to minimize the number of monochromatic k term arithmetic progressions (or k-APs for short). We show how to extend colorings of ℤ_m which avoid nontrivial k-APs to colorings of [n] by an unrolling process. In particular, by using residues to color ℤ_m we produce the best known colorings for minimizing the number of monochromatic k-APs for coloring with r colors for several small values of r and k.

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