= Solution
For a homogeneous ideal gas in a two-dimensional box of area $A$, with dispersion $\epsilon=p^2/(2m)$ and one internal state, the <density of states> is $g(\epsilon)=Am/(2\pi\hbar^2)$. The excited-particle density at <temperature> $T>0$ is
$$
n_{\rm ex}=\frac{m}{2\pi\hbar^2}\int_0^\infty\frac{d\epsilon}{z^{-1}e^{\epsilon/(k_BT)}-1}
=-\frac{mk_BT}{2\pi\hbar^2}\ln(1-z),\qquad z=e^{\mu/(k_BT)}<1.
$$
As $\mu\to0^-$, this diverges. Hence at every fixed positive <temperature> the <excited states> can accommodate any finite areal density with $\mu<0$; they never saturate at a finite critical density. \b[There is no finite-temperature <Bose-Einstein condensation> in the homogeneous two-dimensional ideal gas thermodynamic limit.]
The low-energy logarithmic divergence is the reason, not merely the small number of spatial dimensions. A finite box has a low-energy cutoff and a crossover; a trap changes the <density of states> and can change the conclusion. Condensation at zero <temperature> and an interacting gas's <Berezinskii–Kosterlitz–Thouless transition> are separate issues. The proof assumes the uniform quadratic-dispersion model, rather than every conceivable two-dimensional trapping potential.
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