Intersecting Families of Multisubsets with Rankk

✍ Scribed by Shiquan Wu

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 506 KB
• Volume: 77
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2746

✦ Synopsis

In this note, we obtain the maximum intersecting families of multisubsets with a given rank k. This partly solves an open problem posed by Engel and Frankl. 1997 Academic Press 1. INTRODUCTION Engel and Frankl posed in [2] the following problem on t-intersecting families of multisubsets.

Open Problem. Determine the exact value of the cardinalities of the maximum t-intersecting families of multisubsets with rank k.

The problem remains unsettled for all t. Here, we solve the problem for t=1, i.e., for the case of intersecting families.

Throughout, let n and s be positive integers. Denote (s+1) n = [(x 1 , x 2 , ..., x n ) | 0 x i s, x i is an integer for each i ], which defines the collection of all multisubsets on an n-set X=[1, 2, ..., n] with the repetition x i between 0 and s for each element i. Our aim is to find maximum intersecting families with given rank in (s+1) n . Definition 1. (1) F (s+1) n is called an intersecting family if, for any a, b # F, a i 7b i =min(a i , b i )>0 for some i's. If for any a, b # F, a i 7b i >0 for at least t i's, then F is called a t-intersecting family in (s+1) n .