Geodesic
Anti-van der Waerden numbers of 3-term arithmetic progression
Zhanar Berikkyzy1 ; Alex Shulte1 ; Michael Young1
1Iowa State University
The electronic journal of combinatorics, Tome 24 (2017) no. 2
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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 $7\cdot 3^{m-2}+1 \leq n \leq 21 \cdot 3^{m-2}$, \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*}
DOI : 10.37236/6101
Classification : 11B25, 05D10, 11B50
Mots-clés : arithmetic progression, rainbow coloring, unitary coloring, Behrend construction

Zhanar Berikkyzy  1   ; Alex Shulte  1   ; Michael Young  1

1 Iowa State University 
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     title = {Anti-van der {Waerden} numbers of 3-term arithmetic progression},
     journal = {The electronic journal of combinatorics},
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Zhanar Berikkyzy; Alex Shulte; Michael Young. Anti-van der Waerden numbers of 3-term arithmetic progression. The electronic journal of combinatorics, Tome 24 (2017) no. 2. doi: 10.37236/6101

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