Dominant dimensions,derived equivalences and tilting modules

We show that for every ? > 0 there exist δ > 0 and n₀ ∈ ? such that every 3-uniform hypergraph on n ≥ n₀ vertices with the property that every k-vertex subset, where k ≥ δn, induces at least \(\left( {\frac{1}{2} + \varepsilon } \right)\left( {\begin{array}{*{20}c} k \\ 3 \\ \end{array} } \right)\) edges, contains K₄^? as a subgraph, where K₄^? is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erd?s and Sós. The constant 1/4 is the best possible.