RO-TST 2013 Problem 8

Let $k$ be a positive integer larger than $1$ . Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:(a) any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;(b) any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.