On intersecting hypergraphs

Barry Guiduli, Zoltán Király
1998 Discrete Mathematics  
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 Erdfs and Gyhrfils (1990) (they considered r-wise intersecting and the best t points), and is a generalization of work by Mills (1979) who considered the best single point. These are very hard to calculate in general;
more » ... 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 and M2(12)= 7. To better study this problem, we introduce the concept of fractional matchings and coverings of order 2.
doi:10.1016/s0012-365x(97)00136-2 fatcat:segaq44abjb65mzojpn4xgafda