# Supersaturation For Ramsey-Turán Problems

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 . 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  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 |EE′|≥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  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 , proved by Frankl and Rödl , and later by Sidorenko . Our short proof is based on different ideas and is simpler than these earlier proofs.

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Correspondence to Dhruv Mubayi*.

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* Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship.

† Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.

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Mubayi*, D., Rödl†, V. Supersaturation For Ramsey-Turán Problems. Combinatorica 26, 315–332 (2006). https://doi.org/10.1007/s00493-006-0018-x

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