Abstract
We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specically , 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.
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