### 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 |

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### 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

**Intersection properties of systems of finite sets.** / Deza, M.; Erdős, P.; Frankl, P.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. s3-36, no. 2, pp. 369-384. https://doi.org/10.1112/plms/s3-36.2.369

}

TY - JOUR

T1 - Intersection properties of systems of finite sets

AU - Deza, M.

AU - Erdős, P.

AU - Frankl, P.

PY - 1978

Y1 - 1978

N2 - Let X be a finite set of cardinality n. If L = (l1,…, lr) is a set of nonnegative integers with l12<… 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 > cnr-1 (c = c(k) is a constant depending on k), then (i) there exists an l1-element subset D of X such that D is contained in every member of A (ii) (l2-l1)(l3-l2)… (iii) (Πri=1 (n-li)/(k-li) ≥ A (for n ≥ n0(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 A1,…, Aq+1 there are always two, Ai, Aj, 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’?.

AB - Let X be a finite set of cardinality n. If L = (l1,…, lr) is a set of nonnegative integers with l12<… 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 > cnr-1 (c = c(k) is a constant depending on k), then (i) there exists an l1-element subset D of X such that D is contained in every member of A (ii) (l2-l1)(l3-l2)… (iii) (Πri=1 (n-li)/(k-li) ≥ A (for n ≥ n0(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 A1,…, Aq+1 there are always two, Ai, Aj, 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’?.

UR - http://www.scopus.com/inward/record.url?scp=84959767645&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84959767645&partnerID=8YFLogxK

U2 - 10.1112/plms/s3-36.2.369

DO - 10.1112/plms/s3-36.2.369

M3 - Article

VL - s3-36

SP - 369

EP - 384

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 2

ER -