The Hunt for Observable Signatures of Terrestrial planetary Systems (HOSTS) program on the Large Binocular Telescope Interferometer (LBTI) will survey nearby stars for faint exozodiacal dust (exozodi). This warm circumstellar dust, analogous to the interplanetary dust found in the vicinity of the Earth in our own system, is produced in comet breakups and asteroid collisions. Emission and/or scattered light from the exozodi will be the major source of astrophysical noise for a future space telescope aimed at direct imaging and spectroscopy of terrestrial planets (exo- Earths) around nearby stars. About 20% of nearby field stars have cold dust coming from planetesimals at large distances from the stars (Eiroa et al. 2013, A&A, 555, A11; Siercho et al. 2014, ApJ, 785, 33). Much less is known about exozodi; current detection limits for individual stars are at best ~ 500 times our solar system's level (aka. 500 zodi). LBTI-HOSTS will be the first survey capable of measuring exozodi at the 10 zodi level (3σ). Detections of warm dust will also reveal new information about planetary system architectures and evolution. We will describe the motivation for the survey and progress on target selection, not only the actual stars likely to be observed by such a mission but also those whose observation will enable sensible extrapolations for stars that will not be observed with LBTI. We briefly describe the detection of the debris disk around η Crv, which is the first scientific result from the LBTI coming from the commissioning of the instrument in December 2013, shortly after the first time the fringes were stabilized.