### Abstract

Let p be a positive integer and let Q be a subset of {0, 1,..., p}. Call p sets A_{1},A_{2},...,A_{p} of a ground set X a (p,Q)-system if the number of sets A_{i} containing x is in Q for every x ∈ X. In hypergraph terminology, a (p,Q)-system is a hypergraph with p edges such that each vertex x has degree d(x) ∈ Q. A family of sets ℱ with ground set X is called (p,Q)-free if no p sets of ℱ form a (p,Q)-system on X. We address the Turán-type problem for (p,Q)-systems: f(n,p,Q) is defined as max|ℱ| over all (p,Q)-free families on the ground set [n] = {1,2,...,n}. We study the behavior of f(n,p,Q) when p and Q are fixed (allowing 2^{p+1} choices for Q) while n tends to infinity. The new results of this paper mostly relate to the middle zone where 2^{n-1} ≤ f(n,p,Q) ≤ (1 - c)2^{n} (in this upper bound c depends only on p). This direction was initiated by Paul Erdos who asked about the behavior of f(n,4,{0,3}). In addition, we give a brief survey on results and methods (old and recent) in the low zone (where f(n,p,Q) = o(2^{n})) and in the high zone (where 2^{n} - (2 - c)^{n} < f(n,p,Q)).

Original language | English |
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Pages (from-to) | 385-403 |

Number of pages | 19 |

Journal | Discrete Mathematics |

Volume | 257 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Nov 28 2002 |

Event | Kleitman and Combinatorics: A Celebration - Cambridge, MA, United States Duration: Aug 16 1990 → Aug 18 1990 |

### Keywords

- Degree
- Hypergraph
- Regular

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*257*(2-3), 385-403. https://doi.org/10.1016/S0012-365X(02)00437-5