A hypergraph (finite set system) ℋ is called a bi-Helly family if it satisfies the following property: if any two edges of a subhypergraph ℋl ⊆ ℋ share at least two vertices, then |∩H∈ℋlH|≥2. Solving a problem raised by Voloshin, we prove that the maximum number of edges in a bi-Helly family of given order n and given edge size r≥5 equals (n-2r-2). For r = 3 we show that the maximum equals the Turán number ex(n; script K sign34 - e) (its determination is a famous open problem in extremal hypergraph theory), and for r = 4 we prove the lower and upper bounds n3/26 and n3/20, respectively. Analogous results are presented under the requirement that each pairwise k-intersecting subhypergraph has k universal common elements.
|Number of pages||11|
|Publication status||Published - febr. 28 2000|
|Event||Selected Topics in Discrete Mathematics - Warsaw, Poland|
Duration: aug. 26 1996 → szept. 28 1996
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics