## 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.

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## Additional information

The work leading to this paper has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385. The first author was also supported by DFG within the research training group “Methods for Discrete Structures”. The third author was also supported by the student grant GAUK 601812.

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### Cite this article

Glebov, R., Král’, D. & Volec, J. A problem of Erdős and Sós on 3-graphs.
*Isr. J. Math.* **211, **349–366 (2016). https://doi.org/10.1007/s11856-015-1267-4

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### Keywords

- Convergent Sequence
- Edge Density
- Dense Graph
- Extremal Graph
- Subgraph Frequency