Combinatorica

, Volume 2, Issue 3, pp 289–295

# On Ramsey—Turán type theorems for hypergraphs

• P. Erdős
• Vera T. Sós
Article

## 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.

## AMS subject classification (1980)

05 C 65 05 C 35 05 C 55

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