[Paper Review] Anti-van der Waerden numbers of 3-term arithmetic progressions

Zhanar Berikkyzy, Alex Schulte+1 more|arXiv (Cornell University)|Apr 29, 2016

Limits and Structures in Graph Theory7 references3 citations

TL;DR

This paper resolves the conjecture that the anti-van der Waerden number $\operatorname{aw}([n],3)$ is bounded by $\lceil \log_3 n \rceil + C$ for some constant $C$, by exactly determining $\operatorname{aw}([n],3)$ for all $n$. It proves that $\operatorname{aw}([n],3) = m+2$ if $n = 3^m$, and $m+3$ otherwise, for $7 \cdot 3^{m-2} + 1 \leq n \leq 21 \cdot 3^{m-2}$, confirming the conjecture and establishing the exact threshold for rainbow 3-term arithmetic progressions in exact colorings.

ABSTRACT

The \emph{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 $\lceil \log_3 n ceil + 2 \le aw([n],3) \le \lceil \log_2 n ceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) \le \lceil \log_3 n ceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7\cdot 3^{m-2}+1 \leq n \leq 21 \cdot 3^{m-2}$, \[ aw([n],3)=\left\{\begin{array}{ll} m+2, & \mbox{if $n=3^m$}\\ m+3, & \mbox{otherwise}. \end{array} ight.\]

Motivation & Objective

• To resolve a conjecture by Butler et al. that $\operatorname{aw}([n],3) \leq \lceil \log_3 n \rceil + C$ for some constant $C$.
• To determine the exact value of $\operatorname{aw}([n],3)$ for all $n \geq 2$.
• To establish the relationship between $\operatorname{aw}([n],3)$ and $\operatorname{aw}_u([n],3)$, showing they are equal for all $n$.
• To characterize extremal colorings that avoid rainbow 3-APs using structural and recursive arguments.

Proposed method

• Use of induction on $n$ to prove the exact formula for $\operatorname{aw}([n],3)$, with base cases drawn from Table 1.
• Construction of unitary colorings via modular reduction and coloring extension, using the structure of intervals $[n]$, $[h]$, and $[h-1]$ where $h = \lfloor n/3 \rfloor$.
• Application of Theorem 3 on $\operatorname{aw}(\mathbb{Z}_n,3)$ to bound $\operatorname{aw}(\mathbb{Z}_{2q},3)$ in terms of $\log_3(2q)$, leveraging properties of prime factorization.
• Analysis of special colorings with 8-APs of specific color patterns to derive constraints on $n$ and $q$, leading to bounds on $r$.
• Use of symmetry and reversal of color patterns in intervals to construct rainbow-3-AP-free colorings of $\mathbb{Z}_{2q}$, enabling recursive bounds.
• Proof by contradiction: assuming a coloring with $r$ colors and no rainbow 3-AP leads to a contradiction unless $r \leq f(n) - 1$, thus proving the upper bound.

Experimental results

Research questions

• RQ1 Is there a constant $C$ such that $\operatorname{aw}([n],3) \leq \lceil \log_3 n \rceil + C$ for all $n \geq 3$?
• RQ2 What is the exact value of $\operatorname{aw}([n],3)$ for all $n$?
• RQ3 How do the unitary anti-van der Waerden numbers $\operatorname{aw}_u([n],3)$ relate to $\operatorname{aw}([n],3)$?
• RQ4 What structural properties characterize extremal colorings that avoid rainbow 3-APs?

Key findings

• The conjecture that $\operatorname{aw}([n],3) \leq \lceil \log_3 n \rceil + C$ is confirmed with $C = 3$, and the exact value is determined.
• For $7 \cdot 3^{m-2} + 1 \leq n \leq 21 \cdot 3^{m-2}$, $\operatorname{aw}([n],3) = m+2$ if $n = 3^m$, and $m+3$ otherwise.
• $\operatorname{aw}_u([n],3) = \operatorname{aw}([n],3)$ for all $n \geq 2$, showing unitary colorings achieve the extremal threshold.
• The upper bound $\operatorname{aw}([n],3) \leq f(n)$ is proven via induction and contradiction, using interval reduction and coloring symmetry.
• The analysis shows that if $n \geq 9q$, a 9-AP must contain a rainbow 3-AP, forcing $n \leq 9q - 1$, which restricts possible colorings.
• For $q = 3^i$, $\operatorname{aw}(\mathbb{Z}_{2q},3) \leq \lceil \log_3(2q) \rceil + 2$, and this bound is used to show $r \leq f(n) - 1$ in the inductive step.