  The Experiment to Detect the Global Epoch of Reionization Signature (EDGES)
recently reported a strong 21-cm absorption signal relative to the cosmic
microwave background at $z \sim 18$. While its anomalous amplitude may indicate
new physics, in this work we focus on the timing of the signal, as it alone
provides an important constraint on galaxy formation models. Whereas rest-frame
ultraviolet luminosity functions (UVLFs) over a broad range of redshifts are
well fit by simple models in which galaxy star formation histories track the
assembly of dark matter halos, we find that these same models, with reasonable
assumptions about X-ray production in star-forming galaxies, cannot generate a
narrow absorption trough at $z \sim 18$. If verified, the EDGES signal
therefore requires the fundamental inputs of galaxy formation models to evolve
rapidly at $z \gtrsim 10$. Unless extremely faint sources residing in halos
below the atomic cooling threshold are responsible for the EDGES signal, star
formation in $\sim 10^8$-$10^{10} \ M_{\odot}$ halos must be more efficient
than expected, implying that the faint-end of the UVLF at $M_{\mathrm{UV}}
\lesssim -12$ must steepen at the highest redshifts. This steepening provides a
concrete test for future galaxy surveys with the James Webb Space Telescope and
ongoing efforts in lensed fields, and is required regardless of whether the
amplitude of the EDGES signal is due to new cooling channels or a strong radio
background in the early Universe. However, the radio background solution
requires that galaxies at $z > 15$ emit 1-2 GHz photons with an efficiency
$\sim 10^3$ times greater than local star-forming galaxies, posing a challenge
for models of low-frequency photon production in the early Universe.
