Convex independence and the structure of clone-free multipartite tournaments

We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specifically, we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given convexly independent set. We use this to prove that the Helly number, Radon number, and rank coincide in any clone-free bipartite tournament. We then study the relationship between Helly independence and Radon independence in clone-free multipartite tournaments. We show that if the rank is at least 4 or the Helly number is at least 3, then the Helly number and the Radon number are equal.