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.