### 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 is 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) | 247-249 |

Number of pages | 3 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 10 |

DOIs | |

Publication status | Published - 2001 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

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## Cite this

*Electronic Notes in Discrete Mathematics*,

*10*, 247-249. https://doi.org/10.1016/s1571-0653(04)00403-2