How different can two intersecting families be? (Q2571264)

Summary: To measure the difference between two intersecting families \({\mathcal F}, {\mathcal G} \subseteq 2^{[n]}\) we introduce the quantity \(D({\mathcal F} ,{\mathcal G})=|\{ (F,G):F \in {\mathcal F}\), \(G \in {\mathcal G}\), \(F\cap G =\emptyset \}|\). We prove that if \({\mathcal F}\) is \(k\)-uniform and \({\mathcal G}\) is \(l\)-uniform, then for large enough \(n\) and for any \(i \neq j\) \({\mathcal F}_i=\{F \subseteq [n]: i \in F\), \(|F|=k \}\) and \({\mathcal F}_j=\{F \subseteq [n]: j \in F\), \(|F|=l\}\) form an optimal pair of families (that is \(D({\mathcal F},{\mathcal G}) \leq D({\mathcal F}_i,{\mathcal F}_j)\) for all uniform and intersecting \({\mathcal F}\) and \({\mathcal G}\)), while in the non-uniform case any pair of the form \({\mathcal F}_i=\{F \subseteq [n]: i \in F \}\) and \({\mathcal F}_j=\{F \subseteq [n]: j \in F\}\) is optimal for any \(n\).