On the Independence Numbers of the Cyclic Van der Waerden Hypergraphs

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 $k\ge 3$. 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 $r\ge 2$.

13 pages, added Subsection 3.4 with applications and examples of the main theorem. Comments are welcome!