  We start with a variational approach and derive a set of coupled integral
equations for the bound states of $N$ identical spin-$\uparrow$ fermions and a
single spin-$\downarrow$ fermion in a generic multiband Hubbard Hamiltonian
with an attractive onsite interaction. As an illustration we apply our integral
equations to the one-dimensional sawtooth lattice up to $N \le 3$, i.e., to the
$(3+1)$-body problem, and reveal not only the presence of tetramer states in
this two-band model but also their quasi-flat dispersion when formed in a flat
band. Furthermore, for $N = \{4, 5, \cdots, 10 \}$, our DMRG simulations and
exact diagonalization suggest the presence of larger and larger multimers with
lower and lower binding energies, conceivably without an upper bound on $N$.
These peculiar $(N+1)$-body clusters are in sharp contrast with the exact
results on the single-band linear-chain model where none of the $N \ge 2$
multimers appear. Hence their presence must be taken into account for a proper
description of the many-body phenomena in flat-band systems, e.g., they may
suppress superconductivity especially when there exists a large spin imbalance.
