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Let \( 2^{[m]} \) be ordered by set inclusion, and let \( \mathcal{B} \subseteq 2^{[m]} \) be an antichain. An antichain \( \mathcal{B} \) is called \( k \)-regular (\( k \in \mathbb{N} \)) if for each \( i \in [m] \) there are exactly \( k \) blocks \( B_1, B_2, \ldots, B_k \in \mathcal{B} \) containing \( i \). An antichain is called flat if there exists a positive integer \( l \) such that \( l \leq |B| \leq l+1 \) for all \( B \in \mathcal{B} \), and we call an antichain maximal if the collection of sets \( \mathcal{B} \cup \{B\} \) is not an antichain for all \( B \notin \mathcal{B} \). We call a maximal \( k \)-regular antichain \( \mathcal{B} \subseteq \binom{[m]}{2} \cup \binom{[m]}{3} \) a \( (k,m) \)-MFRAC. In this paper we analyze \( (k,m) \)-MFRACs in the cases \( m \leq 7 \), \( k = m \), \( k = m-1 \), and \( k = m-2 \). We provide some constructions, give necessary conditions for existence, and mention some open problems.