Largest size and union of Helly families

AbstractA finite set system (hypergraph)H is said to have the Helly property if the members of each intersecting subsystem H′ of H share an element (i.e. H∩H′≠øFS for all H,H′∈H′ implies ∩H∈H′ H≠ø); H is k-uniform if |H|=k for all H∈H. In a previous paper, we con jectured that the union of t k-uniform Helly families (k⩾3) on the same n-element set can have at most ∑ti=1(n−ik−1) members. Here we prove this conjecture for every k and every t ⩽2k−2, for n sufficiently large with respect to k.The main tool is a result (an analogue of the Hilton-Milner theorem on intersecting k-uniform set systems) stating that if the sets in a k-uniform Helly family on n points have an empty intersection, then for large n, |H|⩽(n−k−1k−1)+(n−2k−2)+1, and the set system attaining equality is uniq ue.We also show that ∑ti=1(n−ik−1) is a (best possible) upper bound for the union of t intersecting k-uniform set systems on the same n-element set, whenever n⩾(t+2k−1)(k−1). This result strengthens a particular case of the Hajnal-Rothschild theorem