On the Chudnovsky-Seymour-Sullivan conjecture on cycles in triangle-free digraphs

For a simple digraph G without directed triangles or digons, let \beta(G) be the size of the smallest subset X of E(G) such that G\X has no directed cycles, and let \gamma(G) be the number of unordered pairs of nonadjacent vertices in G. In 2008, Chudnovsky, Seymour, and Sullivan showed that \beta(G) \leq \gamma(G) and conjectured that \beta(G) \leq \gamma(G)/2. Recently, Dunkum, Hamburger, and Por proved that \beta(G)\leq .88\gamma(G). In this note, we prove that \beta(G) \leq .8616 \gamma(G).