### Abstract

Denote by G^{(r)}(n;k) an r-graph of n vertices and k r-tuples. Turán's classical problem states: Determine the smallest integer f{hook}(n;r,l) so that every G^{(r)}(n;f{hook}(n;r,l)) contains a K^{(r)}(l). Turán determined f{hook}(n;r,l) for r = 2, but nothing is known for r > 2. Put lim_{n=-} f{hook}(n;r,l) ( n r)= c_{r,l}. The values of c_{r,l} are not known for r > 2. I prove that to every e > 0 and integer t there is an n_{0} = n_{0}(t,ε{lunate}) so that every G(^{r})(n;[(c_{r,l}+ε{lunate})( n R)]) has lt vertices x_{t}(^{j}), l≤i≤t,l≤j≤l, so that all the r-tuples {X_{i1}^{(j1)},...X_{ir}^{(jr)}}, l≤i_{s}≤t,l≤j_{l}< ... <j_{r}≤l, occur in our G^{(r)}. Several unsolved problems are posed.

Original language | English |
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Pages (from-to) | 1-6 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 1971 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics