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 [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 |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 [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.

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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).

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Mathematics Subject Classification (2000):

  • 05C35
  • 05C65
  • 05D05