Anti-van der Waerden numbers of 3-term arithmetic progressions

2016/04/29 by Berikkyzy, Zhanar, Schulte, Alex, Young, Michael · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.08819

Abstract

The anti-van der Waerden number, denoted by aw([n],k), is the smallest r such that every exact r-coloring of [n] contains a rainbow k-term arithmetic progression. Butler et. al. showed that \lceil log₃ n \rceil + 2 ≤ aw([n],3) ≤ \lceil log₂ n \rceil + 1, and conjectured that there exists a constant C such that aw([n],3) ≤ \lceil log₃ n \rceil + C. In this paper, we show this conjecture is true by determining aw([n],3) for all n. We prove that for 7⋅ 3^m-2+1 ≤ n ≤ 21 ⋅ 3^m-2, aw([n],3)=\m+2, · amp; if n=3^m
m+3, · amp; otherwise..

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