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From the list of 2321 transiting planet candidates announced by the KeplerMission, we select seven targets with favorable properties for the capacity todynamically maintain an exomoon and present a detectable signal. These sevencandidates were identified through our automatic target selection (TSA)algorithm and target selection prioritization (TSP) filtering, whereby weexcluded systems exhibiting significant time-correlated noise and focussed onthose with a single transiting planet candidate of radius less than 6 Earthradii. We find no compelling evidence for an exomoon around any of the sevenKOIs but constrain the satellite-to-planet mass ratios for each. For four ofthe seven KOIs, we estimate a 95% upper quantile of M_S/M_P<0.04, which giventhe radii of the candidates, likely probes down to sub-Earth masses. We alsoderive precise transit times and durations for each candidate and find noevidence for dynamical variations in any of the KOIs. With just a few systemsanalyzed thus far in the in-going HEK project, projections on eta-moon would bepremature, but a high frequency of large moons aroundSuper-Earths/Mini-Neptunes would appear to be incommensurable with our resultsso far.

The Hunt for Exomoons with Kepler (HEK): II. Analysis of Seven Viable Satellite-Hosting Planet Candidates