Let \( C^{{{\left( {2k} \right)}}}_{r} \) be the 2*k*-uniform hypergraph obtained by letting *P*_{1}, . . .,*P*_{r} be pairwise disjoint sets of size *k* and taking as edges all sets *P*_{
i
}∪*P*_{
j
} with *i* ≠ *j*. This can be thought of as the ‘*k*-expansion’ of the complete graph *K*_{
r
}: each vertex has been replaced with a set of size *k*. An example of a hypergraph with vertex set *V* that does not contain \( C^{{{\left( {2k} \right)}}}_{3} \) can be obtained by partitioning *V* = *V*1 ∪*V*_{2} and taking as edges all sets of size 2*k* that intersect each of *V*_{1} and *V*_{2} in an odd number of elements. Let \( {\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n} \) denote a hypergraph on *n* vertices obtained by this construction that has as many edges as possible. For *n* sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on *n* vertices that contains no \( C^{{{\left( {2k} \right)}}}_{3} \) has at most as many edges as \( {\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n} \).

Sidorenko has given an upper bound of \( \frac{{r - 2}} {{r - 1}} \) for the Tur´an density of \( C^{{{\left( {2k} \right)}}}_{r} \) for any *r*, and a construction establishing a matching lower bound when *r* is of the form 2^{p}+1. In this paper we also show that when *r*=2^{p}+1, any \( C^{{{\left( 4 \right)}}}_{r} \)-free hypergraph of density \( \frac{{r - 2}} {{r - 1}} - o{\left( 1 \right)} \) looks approximately like Sidorenko’s construction. On the other hand, when *r* is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of \( C^{{{\left( 4 \right)}}}_{r} \) to \( \frac{{r - 2}} {{r - 1}} - c{\left( r \right)} \), where *c*(*r*) is a constant depending only on *r*.

The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.

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* Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.

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Keevash, P., Sudakov*, B. On A Hypergraph Turán Problem Of Frankl.
*Combinatorica* **25, **673–706 (2005). https://doi.org/10.1007/s00493-005-0042-2

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*Mathematics Subject Classification (2000):*

- 05C35
- 05C65
- 05D05
- 05E35