Largest size and union of Helly families (Q1322238)

A \(k\)-uniform hypergraph is a hypergraph in which all edges have precisely \(k\) vertices. A hypergraph is called intersecting if and only if any two of its edges share a vertex. A \(k\)-uniform hypergraph is called a Helly family if each of its intersecting subhypergraphs has the property that the intersection of all its edges is not empty. The author's conjecture that the union of \(t\) \(k\)-uniform Helly families on sets of \(n\) elements can have at most \(\sum^ t_{i=1} {n-i\choose k-1}\) members is proved for every \(k\) and every \(t= 2k-2\) if \(n\) is sufficiently large with respect to \(k\), and \(\sum^ t_{i=1} {n- i\choose k-1}\) is a best upper bound whenever \(n\geq (t+ 2k- 1)(k- 1)\).