Extremal bi-Helly families Original Research Article

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 H′⊆H share at least two vertices, then |⋂H∈H′H|⩾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;L43−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.