For an *l*-graph \(
{\user1{\mathcal{G}}}
\), the Turán number \(
{\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}
\) is the maximum number of edges in an *n*-vertex *l*-graph \(
{\user1{\mathcal{H}}}
\) containing no copy of \(
{\user1{\mathcal{G}}}
\). The limit \(
\pi {\left( {\user1{\mathcal{G}}} \right)} = \lim _{{n \to \infty }} {\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}/{\left( {^{n}_{l} } \right)}
\) is known to exist [8]. The Ramsey–Turán density \(
p{\left( {\user1{\mathcal{G}}} \right)}
\) is defined similarly to \(
\pi {\left( {\user1{\mathcal{G}}} \right)}
\) except that we restrict to only those \(
{\user1{\mathcal{H}}}
\) with independence number *o*(*n*). A result of Erdős and Sós [3] states that \(
\pi {\left( {\user1{\mathcal{G}}} \right)} = p{\left( {\user1{\mathcal{G}}} \right)}
\) as long as for every edge *E* of \(
{\user1{\mathcal{G}}}
\) there is another edge *E*′of \(
{\user1{\mathcal{G}}}
\) for which |*E*∩*E*′|≥2. Therefore a natural question is whether there exists \(
{\user1{\mathcal{G}}}
\) for which \(
p{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}
\).

Another variant \(
\ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)}
\) proposed in [3] requires the stronger condition that every set of vertices of \(
{\user1{\mathcal{H}}}
\) of size at least *εn* (0<ε<1) has density bounded below by some threshold. By definition, \(
\ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)} \leqslant p{\left( {\user1{\mathcal{G}}} \right)} \leqslant \pi {\left( {\user1{\mathcal{G}}} \right)}
\) for every \(
{\user1{\mathcal{G}}}
\). However, even \(
\ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}
\) is not known for very many *l*-graphs \(
{\user1{\mathcal{G}}}
\) when *l*>2.

We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each *l*≥3, infinitely many l-graphs \(
{\user1{\mathcal{G}}}
\) for which \(
0 < \ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}
\).

We also prove that the 3-graph \( {\user1{\mathcal{G}}} \) with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies \( 0 < p{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)} \). The existence of a hypergraph \( {\user1{\mathcal{H}}} \) satisfying \( 0 < p{\left( {\user1{\mathcal{H}}} \right)} < \pi {\left( {\user1{\mathcal{H}}} \right)} \) was conjectured by Erdős and Sós [3], proved by Frankl and Rödl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs.

## *Mathematics Subject Classification (2000):*

05C35 05C65 05D05 ## Preview

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