Maximal s-Wise t-Intersecting Families of Sets: Kernels, Generating Sets, and Enumeration
✍ Scribed by Lucia Moura
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 239 KB
- Volume
- 87
- Category
- Article
- ISSN
- 0097-3165
No coin nor oath required. For personal study only.
✦ Synopsis
For fixed s, n, k, and t, let I s (n, k, t) denote the set of all such families. A family A # I s (n, k, t) is said to be maximal if it is not properly contained in any other family in I s (n, k, t). We show that for fixed s, k, t, there is an integer n 0 =n 0 (k, s, t), for which the maximal families in I s (n 0 , k, t) completely determine the maximal families in I s (n, k, t), for all n n 0 . We give a construction for maximal families in I s (n+1, k+1, t+1) based on those in I s (n, k, t). Finally, for s=2, we classify the maximal families for k=t+1, n t+2, t 1, and for k=t+2, n t+6, t 1. The concepts of kernels and generating sets of a family of subsets play an important role in this work.