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- W2962762798 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 $lceil log_3 n rceil + 2 le aw([n],3) le lceil log_2 n rceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) le lceil log_3 n rceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7cdot 3^{m-2}+1 leq n leq 21 cdot 3^{m-2}$, begin{equation*} aw([n],3)=left{begin{array}{ll} m+2, & mbox{if $n=3^m$} m+3, & mbox{otherwise}. end{array}right. end{equation*}" @default.
- W2962762798 created "2019-07-30" @default.
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- W2962762798 date "2017-06-16" @default.
- W2962762798 modified "2023-09-26" @default.
- W2962762798 title "Anti-van der Waerden Numbers of 3-Term Arithmetic Progression" @default.
- W2962762798 doi "https://doi.org/10.37236/6101" @default.
- W2962762798 hasPublicationYear "2017" @default.
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