Most probably intersecting hypergraphs (Q2341073)

Summary: The celebrated Erdős-Ko-Rado theorem shows that for \(n \geq 2k\) the largest intersecting \(k\)-uniform set family on \([n]\) has size \(\binom{n-1}{k-1}\). It is natural to ask how far from intersecting larger set families must be. \textit{G. O. H. Katona} et al. [Comb. Probab. Comput. 21, No. 1--2, 219--227 (2012; Zbl 1241.05144)] introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.{ }We consider the most probably intersecting problem for \(k\)-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.