# Extremal set systems with restricted k-wise intersections

Z. Füredi, Benny Sudakov

Research output: Contribution to journalArticle

13 Citations (Scopus)

### Abstract

A large variety of problems and results in Extremal Set Theory deal with estimates on the size of a family of sets with some restrictions on the intersections of its members. Notable examples of such results, among others, are the celebrated theorems of Fischer, Ray-Chaudhuri-Wilson and Frankl-Wilson on set systems with restricted pairwise intersections. These also can be considered as estimates on binary codes with given distances. In this paper we obtain the following extension of some of these results when the restrictions apply to k-wise intersections, for k > 2. Let L be a subset of non-negative integers of size s and let k > 2. A family F of subsets of an n-element set is called k-wise L-intersecting if the cardinality of the intersection of any k distinct members in F belongs to L. We prove that, for any fixed k and s and sufficiently large n, the size of every k-wise L-intersecting family is bounded by A figure is presented. This result is asymptotically best possible. In addition, we show that for an extremal k-wise L-intersecting family, L consists of s consecutive integers. Our proof combines tools from linear algebra with some combinatorial arguments.

Original language English 143-159 17 Journal of Combinatorial Theory, Series A 105 1 https://doi.org/10.1016/j.jcta.2003.10.008 Published - Jan 2004

### Fingerprint

Set Systems
Intersection
Intersecting Family
Extremal Set Theory
Restriction
Combinatorial argument
Binary codes
Linear algebra
Integer
Subset
Binary Code
Set theory
Estimate
Half line
Consecutive
Pairwise
Cardinality
Figure
Non-negative
Distinct

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Theoretical Computer Science

### Cite this

Extremal set systems with restricted k-wise intersections. / Füredi, Z.; Sudakov, Benny.

In: Journal of Combinatorial Theory, Series A, Vol. 105, No. 1, 01.2004, p. 143-159.

Research output: Contribution to journalArticle

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