Let n≥2 be an integer, let S be a set of n elements, and let Ai,1≤i≤m, be distinct subsets of S of size at least 2 such that A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ . Show that m \leq 2^{n\minus{}1}\minus{}1.
P. Erdos combinatorics proposedcombinatorics