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We present ultra-deep, matched-resolution Karl G. Jansky Very Large Array(VLA) observations at 10 and $3$ GHz in the COSMOS field: the COSMOS-XS survey.The final 10 and $3$ GHz images cover $\sim16\rm{arcmin}^{2}$ and$\sim180\rm{arcmin}^{2}$ and reach median rms values of$0.41\mu\rm{Jy\,beam}^{-1}$ and $0.53\mu\rm{Jy\,beam}^{-1}$, respectively. Bothimages have an angular resolution of $\sim 2.0''$. To fully account for thespectral shape and resolution variations across the broad bands, we image alldata with a multi-scale, multi-frequency synthesis algorithm. We present sourcecatalogs for the 10 and $3$ GHz image with 91 and 1498 sources, respectively,above a peak brightness threshold of $5\sigma$. We present source counts withcompleteness corrections included that are computed via Monte Carlosimulations. Our corrected radio counts at $3$ GHz with direct detections downto $\sim2.8\mu$Jy are consistent within the uncertainties with other results at3 and 1.4 GHz, but extend to fainter flux densities than previous directdetections. The ultra-faint $3$ GHz number counts are found to exceed thecounts predicted by the semi-empirical radio sky simulations developed in theframework of the SKA Simulated Skies project, consistent with previous P(D)analyses. Our measured source counts suggest a steeper luminosity functionevolution for these faint star-forming sources. The semi-empirical Tiered RadioExtragalactic Continuum Simulation (T-RECS) predicts this steeper evolution andis in better agreement with our results. The $10$ GHz radio number counts alsoagree with the counts predicted by the T-RECS simulation within the expectedvariations from cosmic variance. In summary, the multi-band, matched-resolutionCOSMOS-XS survey in the well-studied COSMOS field provides a high-resolutionview of the ultra-faint radio sky that can help guide next generation radiofacilities.

An Ultra-deep Multi-band VLA Survey of the Faint Radio Sky (COSMOS-XS): Source Catalog and Number Counts