Biject the set of $k$-element sets of positive integers with the set of positive integers to label the former $S_1, S_2, \dots, S_n, \dots$. For every positive integer $m$, set $A_m = \{n : m \in S_n\}$.

If $m$ and $m'$ are distinct positive integers, there exist distinct positive integers $n$ and $n'$ such that $m \in S_n$ and $m' \in S_{n'}$. Consequently, $n \in A_m \setminus A_{m'}$ and $n' \in A_{m'} \setminus A_m$; in particular, $A_m \neq A_{m'}$, so the $\mathcal{A}$'s form an infinite set $\mathcal{A}$.

Next, if $m_1, m_2, \dots, m_k$ are distinct positive integers, then $A_{m_1} \cap A_{m_2} \cap \dots \cap A_{m_k} = \{n\}$, where $n$ is the index of the label of the set $\{m_1, m_2, \dots, m_k\}$ in the list $S_1, S_2, \dots$. Consequently, $\mathcal{A}$ satisfies (a).

Finally, if $m_1, m_2, \dots, m_k, m_{k+1}$ are distinct positive integers, then $\{m_1, m_2, \dots, m_k\}$ and $\{m_2, \dots, m_k, m_{k+1}\}$ have different labels in the list $S_1, S_2, \dots$, so $A_{m_1} \cap A_{m_2} \cap \dots \cap A_{m_k} \cap A_{m_{k+1}}$ is empty. Consequently, $\mathcal{A}$ satisfies (b).