Topology of acyclic complexes of tournaments and coloring

Abstract

We prove that the acyclic complex $\mathrm{Acy }(T)$ of any trisectionable tournament $T$ is homotopy equivalent to a wedge of spheres, and show that there exists a fix number $0<\epsilon <1$ such that if $T$ is a trisectionable tournament and $d$ is the highest dimension of a sphere occurring in such a decomposition for $\mathrm{Acy }(T)$, then the (acyclic) chromatic number of $T$ satisfies $\chi (T)\leqslant c(d+1)^{\frac{1}{\epsilon }-1}-1$ for some $1.62<c\leqslant 2$, and by way of an example, we verify that the provided upper bound is tight.