Convex invariants in multipartite tournaments

In the study of convexity spaces, the most common convex invariants are based on notions of independence with respect to taking convex hulls. In [D.B. Parker, R.F. Westhoff and M.J. Wolf, Discuss. Math. Graph Theory 29 (2009), 51–69], H-independence, R-independence and convex independence were studied to prove results about the Helly number, Radon number and rank of a clone-free multipartite tournament under 2-path convexity. In this paper, we extend many of these results to general multipartite tournaments. In particular, we determine conditions under which a convexly independent set of vertices is Rand/or H-independent. We also investigate conditions under which an R-independent set is Hindependent.