An application of flag algebras to a problem of Erdős and Sós

We show that for every ε > 0 there exist δ > 0 and n 0 ∈ N such that every 3-uniform hypergraph on n ⩾ n 0 vertices with the property that every k-vertex subset, where k ⩾ δ n , induces at least ( 1 4 + ϵ ) ( k 3 ) edges, contains K 4 − as a subgraph, where K 4 − is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.