# A problem of Erdős and Sós on 3-graphs

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## Abstract

We show that for every *ɛ* > 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* ≥ *δn*, induces at least \(\left( {\frac{1}
{2} + \varepsilon } \right)\left( {\begin{array}{*{20}c}
k \\
3 \\
\end{array} } \right)\) 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.

## Keywords

Convergent Sequence Edge Density Dense Graph Extremal Graph Subgraph Frequency
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