Examples of topologically highly chromatic graphs with locally small chromatic number

Kierstead, Szemer'edi, and Trotter showed that a graph with at most $\lfloor r/(2n)\rfloor^n$verticessuchthateachballofradius$r$initis$c$-colorableshouldhavechromaticnumberatmost$n(c-1)+1$. We show that this estimate is sharp in $r$. Namely, for every $n$, $r$, and $c$ we construct a graph $G$ containing $O((2rc{)}^{n-1}c)$ vertices such that $\chi(G)\geq n(c-1)+1$,althougheachballofradius$r$in$G$is$c$-colorable. The core idea is the construction of a graph whose neighborhood complex is homotopy equivalent to the join of neighborhood complexes of two given graphs.