Topology of acyclic complexes of tournaments and coloring

We prove that the acyclic complex of any trisectionable tournament is homotopy equivalent to a wedge of spheres, and show that there exists a fix number such that if is a trisectionable tournament and is the highest dimension of a sphere occurring in such a decomposition for , then the (acyclic) chromatic number of satisfies for some , and by way of an example, we verify that the provided upper bound is tight.