Problem 2

2. Does there exist an infinite collection of sets such that the intersection of every two distinct sets in the collection is nonempty, but the intersection of every three distinct sets in the collection is empty?

Proof: The answer is yes; the concept behind constructing a counterexample is in adducing an infinite collection of sets such that the intersection of any two distinct sets results in a singleton set whose element is independently identifiable as belonging to precisely those two sets.

For all $i∈ℕ^+$ comprehend from $\mathcal P(ℕ)$ the set $A_i=\{\{i,j\}~|~j∈ℕ\}$. For distinct $i,j$ we see $A_i∩A_j = \{\{i,j\}\}$ yet this $\{i,j\}∉A_k$ for any third distinct $k$.