On intersecting hypergraphs

We investigate the following question: “Given an intersecting multi-hypergraph on n points, what fraction of edges must be covered by any of the best 2 points?” (Here “best” means that together they cover the most.) We call this M2(n). This is a special case of a question asked by Erdős and Gyárfás [1] (they considered r–wise intersecting and the best t points), and is a generalization of work by Mills [6], who considered the best single point. These are very hard to calculate in general; we show that determining M2(q 2 + q+1) proves the existence or nonexistence of a projective plane of order q. If such a projective plane exists, we conjecture that M2(q 2+q+2) = M2(q 2+q+1). We further show that M2(q 2 + q + 3) < M2(q 2 + q + 1) and conjecture that M2(n+ 2) < M2(n) for all n. We determine the specific values for n ≤ 10. In particular we have the surprising result that M2(7) = M2(8), leading to the conjecture made above. We further conjecture that M2(11) = 5/8 and M2(12) = 7/12. To better study this problem, we introduce the concept of fractional matchings and coverings of order 2.