# 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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## References

1. [1]

B. Bollobás andP. Erdős, On a Ramsey—Turán type problem,J. Comb. Th. Ser. B 21 (1976) 166–168.

2. [2]

P. Erdős, A. Hajnal, V. T. Sós andE. Szemerédi, More results on Ramsey—Turán type problems,Combinatorica,3 (1) (1983).

3. [3]

P. Erdős andA. Hajnal, On chromatic number of graphs and set systems,Acta Math. Acad. Sci. Hungar. 17 (1966) 61–99.

4. [4]

P. Erdős, On extremal problems of graphs and generalized graphs,Israel J. Math. 2 (1964) 183–190.

5. [5]

P. Erdős andV. T. Sós, Some remarks on Ramsey’s and Turán’s theorem.Comb. Theory and Appl. (P. Erdős et al. eds.)Math. Coll. Soc. J. Bolyai 4 Balatonfüred (1969) 395–404.

6. [6]

P. Erdős andM. H. Stone, On the structure of linear graphs,Bull. Amer. Math. Soc. 52 (1946) 1087–1091.

7. [7]

V. T. Sós, On extremal problems in graph theoryProc. Calgary Internat. Conf. on Comb. Structures (1969) 407–410.

8. [8]

E. Szemerédi, On graphs containing no complete subgraph with 4 vertices (in Hungarian),Mat. Lapok 23 (1972) 111–116.

9. [9]

P. Turán. Eine Extremalaufgabe aus der Graphentheorie (in Hungarian),Mat. Fiz. Lapok 48 (1941) 436–452.see also: On the theory of graphs,Colloquium Math. 3 (1954), 19–30.

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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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