A Probabilistic Threshold for Monochromatic Arithmetic Progressions

2012/06/13 by Aaron Robertson, Robertson, Aaron
Mathematics · #05D10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1206.2885

openalex publication_date 2012/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that √(k)⋅ r^k/2 is a threshold interval length where, under mild conditions, almost every r-coloring of an interval of longer length contains a monochromatic k-term arithmetic progression, while almost no r-coloring of an interval of shorter length contains a monochromatic k-term arithmetic progression.

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