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arXiv · 2509.07926

On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs

Benjamin Liber
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Abstract

Building upon the work of Berglund (2018), we establish a method for constructing subsets $B \subseteq \mathbb{Z}_{mk}$ such that $B$ does not contain any $k$-term cyclic arithmetic progressions mod $mk$, where $m,k \in \mathbb{Z}^+$ with $k \geq 3$. This construction thereby provides concrete lower bounds for the maximum size of such subsets. Additionally, it allows us to tightly bound specific chromatic numbers $\chi(mk,k)$ of $\mathbb{Z}_{mk}$ and helps increase the lower bounds of certain cyclic Van der Waerden numbers $W_{c}(k,r)$, originally introduced by Burkert and Johnson (2011) as a way of bounding the standard Van der Waerden numbers $W(k,r)$ from below for $r \geq 2$.

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Benjamin Liber. 2025-09-09. On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs. https://arxiv.org/abs/2509.07926

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