The EDGES experiment recently announced evidence for a broad absorption feature in the sky-averaged radio spectrum around 78 MHz, as may result from absorption in the 21 cm line by neutral hydrogen at z~15-20. If confirmed, one implication is that the spin temperature of the 21 cm line is coupled to the gas temperature by z=20. The known mechanism for accomplishing this is the Wouthuysen-Field effect, whereby Lyman-alpha photons scatter in the intergalactic medium (IGM) and impact the hyperfine level populations. This suggests that early star formation had already produced a copious Lyman-alpha background by z=20, and strongly constrains models in which the linear matter power spectrum is suppressed on small-scales, since halo and star formation are delayed in such scenarios. Here we consider the case that the dark matter consists of ultra-light axions with macroscopic de Broglie wavelengths (fuzzy dark matter, FDM). We assume that star formation tracks halo formation and adopt two simple models from the current literature for the halo mass function in FDM. We further suppose that the fraction of halo baryons which form stars is less than a conservative upper limit of $f_star leq 0.05$, and that ~10^4 Lyman-alpha to Lyman-limit photons are produced per stellar baryon. We find that the requirement that the 21 cm spin temperature is coupled to the gas temperature by $z=20$ places a lower-limit on the FDM particle mass of $m_a geq 5 times 10^{-21} {rm eV}$. The constraint is insensitive to the precise minimum mass of halos where stars form. As the global 21 cm measurements are refined, the coupling redshift could change and we quantify how the FDM constraint would be modified. A rough translation of the FDM mass bound to a thermal relic warm dark matter (WDM) mass bound is also provided.
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