### Abstract

Let X be a finite set of cardinality n. If L = (l_{1},…, l_{r}) is a set of nonnegative integers with l_{1}2<… <l_{r}, and & is a natural number, then by an (n, L, k)-system we mean a collection of k-element subsets of X such that the intersection of any two different sets has cardinality belonging to L. We prove that if A is an (n, L, k)-system, with A > cn^{r-1} (c = c(k) is a constant depending on k), then (i) there exists an l_{1}-element subset D of X such that D is contained in every member of A (ii) (l_{2}-l_{1})(l_{3}-l_{2})… (iii) (Π^{r}_{i=1} (n-l_{i})/(k-l_{i}) ≥ A (for n ≥ n_{0}(k)). Parts of the results are generalized for the following cases: (a) we consider t-wise intersections, where t ≥ 2; (b) the condition A= k is replaced by A K where K is a set of integers; (c) the intersection condition is replaced by the following: Among q +1 different members A_{1},…, A_{q+1} there are always two, A_{i}, A_{j}, such that e Ai ∩ Aj ∈ L. We consider some related problems. An open question: let L’ ⊂ L do there exist an [n, L, k)-system of maximal cardinality (A) and an (n, L’, k)-system of maximal cardinality (A’) such that A A’?.

Original language | English |
---|---|

Pages (from-to) | 369-384 |

Number of pages | 16 |

Journal | Proceedings of the London Mathematical Society |

Volume | s3-36 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1978 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the London Mathematical Society*,

*s3-36*(2), 369-384. https://doi.org/10.1112/plms/s3-36.2.369