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Combinatorica

, Volume 26, Issue 3, pp 315–332 | Cite as

Supersaturation For Ramsey-Turán Problems

  • Dhruv Mubayi*Email author
  • Vojtěch Rödl†
Original Paper
  • 59 Downloads

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.

Mathematics Subject Classification (2000):

05C35 05C65 05D05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of IllinoisChicago, IL 60607-7045USA
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlanta, GA 30322USA

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