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

We show that for every ε>0 there exist δ>0 and n0∈N such that every 3-uniform hypergraph on n⩾n0 vertices with the property that every k-vertex subset, where k⩾δn, induces at least edges, contains as a subgraph, where 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.