Turán number of 3-free strong digraphs with out-degree restriction (Q2127631)

A digraph is 3-free if it contains no directed cycle of length less than 4. The authors refine an extremal result in [\textit{J. C. Bermond} et al., J. Graph Theory 4, 337--341 (1980; Zbl 0412.05048)] which implies that the Turán number of strong simple digraphs \(D=(V,A)\) with \(|V|=n\) containing no directed cycle of length less than 4 is \(\binom{n-1}{2}+1\). They continue a study they began in [\textit{B. Chen} and \textit{A. Chang}, Graphs Comb. 37, No. 6, 2535--2554 (2021; Zbl 1479.05128)], defining, as there, \(\Phi_n(\xi,\gamma)\) to be the family of 3-free \(D=(V,A)\) where \(|V|=n\), in which the minimum out-degree and minimum in-degree are respectively at least \(\xi>0\) and at least \(\gamma>0\); \(\phi_n(\xi,\gamma)\) is the maximum of \(|A|\) for \(D\in\Phi_n(\xi,\gamma)\). In that earlier paper, the present authors studied extremal digraphs, determined the value of \(\phi_n(1,1)\) to be \(\binom{n-1}{2}+1\), and investigated bounds for \(\phi_n(2,1)\). In the present paper, they prove Theorem 1.6: \(\binom{n-1}{2}-2\le\phi_n(2,1)\le\binom{n-1}{2}-1\); they prove specifically that \(\phi_7(2,1)=\binom{7-1}{2}-1=14\), \(\phi_8(2,1)=\binom{8-1}{2}-1=20\), and \(\phi_9(2,1)=\binom{9-1}{2}-2=26\).