  The number density of small dark matter (DM) halos hosting faint
high-redshift galaxies is sensitive to the DM free-streaming properties.
However, constraining these DM properties is complicated by degeneracies with
the uncertain baryonic physics governing star formation. In this work, we use a
flexible astrophysical model and a Bayesian inference framework to analyse
ultra-violet (UV) luminosity functions (LFs) at z=6-8. We vary the complexity
of the galaxy model (single vs double power law for the stellar -- halo mass
relation) as well as the matter power spectrum (cold DM vs thermal relic warm
DM), comparing their Bayesian evidences. Adopting a conservatively wide prior
range for the WDM particle mass, we show that the UV LFs at z=6-8 only weakly
favour CDM over WDM. We find that particle masses of $\lesssim$2 keV are
rejected at a 95% credible level in all models that have a WDM-like power
spectrum cutoff. This bound should increase to ~2.5 keV with the James Webb
Space Telescope (JWST).
