Turán problems in directed and mixed graphs

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We investigate natural Turán problems for directed graphs (digraphs) and mixed graphs, generalizations of graphs where edges can be either directed or undirected. In both of these situations, far less is known than in the classical Turán setting. We characterize the Turán numbers of several natural families of sparse digraphs that highlight that directed path lengths play a weaker, but analogous role to the chromatic number in the directed setting. In the mixed graph setting, we study a natural Turán density coefficient, establishing an analogue of the Erdős-Stone-Simonovits theorem and showing that Turán density coefficients can be irrational, but are always algebraic.

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