  We determine the viability of 4-layered Aurivillius phases to exhibit
long-range magnetic order above room temperature. We use Monte Carlo
simulations to calculate transition temperatures for an effective Heisenberg
model containing a minimal set of required couplings. The magnitude of the
corresponding coupling constants has been determined previously from electronic
structure calculations for $\mathrm{Bi_5FeTi_3O_{15}}$, for which we obtain a
transition temperature far below room temperature. We analyze the role of
further neighbor interactions within our Heisenberg model, in particular that
of the second-nearest-neighbor coupling within the perovskite-like layers of
the Aurivillius structure, as well as that of the weak inter-layer coupling, in
order to identify the main bottleneck for achieving higher magnetic transition
temperatures. Based on our findings, we show that the most promising strategy
to obtain magnetic order at higher temperatures is to increase the
concentration of magnetic cations within the perovskite-like layers, and we
propose candidate compounds where magnetic order could be achieved above room
temperature.
