On Ramsey—Turán type theorems for hypergraphs

Abstract

LetH r be anr-uniform hypergraph. Letg=g(n;H r) be the minimal integer so that anyr-uniform hypergraph onn vertices and more thang edges contains a subgraph isomorphic toH r. Lete =f(n;H r,εn) denote the minimal integer such that everyr-uniform hypergraph onn vertices with more thane edges and with no independent set ofεn vertices contains a subgraph isomorphic toH r.

We show that ifr>2 andH r is e.g. a complete graph then

$$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r )$$

while for someH r with\(\mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} g(n;H^r ) \ne 0\)

$$\mathop {\lim }\limits_{\varepsilon \to 0} \mathop {\lim }\limits_{n \to \infty } \left( {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right)^{ - 1} f(n;H^r ,\varepsilon n) = 0$$

. This is in strong contrast with the situation in caser=2. Some other theorems and many unsolved problems are stated.

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Dedicated to Tibor Gallai on his seventieth birthday

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Erdős, P., Sós, V.T. On Ramsey—Turán type theorems for hypergraphs. Combinatorica 2, 289–295 (1982). https://doi.org/10.1007/BF02579235

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AMS subject classification (1980)

  • 05 C 65
  • 05 C 35
  • 05 C 55