The minimum size of 3-graphs without a 4-set spanning no or exactly three edges

Let GiGi be the (unique) 3-graph with 4 vertices and ii edges. Razborov [A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010) 946–963] determined asymptotically the minimum size of a 33-graph on nn vertices having neither G0G0 nor G3G3 as an induced subgraph. Here we obtain the corresponding stability result, determine the extremal function exactly, and describe all extremal hypergraphs for n≥n0n≥n0. It follows that any sequence of almost extremal hypergraphs converges, which answers in the affirmative a question posed by Razborov.