  Using the EVEREST photometry pipeline, we have identified 74 candidate
ultra-short-period planets (orbital period P<1 d) in the first half of the K2
data (Campaigns 0-8 and 10). Of these, 33 candidates have not previously been
reported. A systematic search for additional transiting planets found 13 new
multi-planet systems, doubling the number known and representing a third (32%)
of USPs. We also identified 30 companions, which have periods from 1.4 to 31
days (median 5.5 d). A third (36 of 104) of the candidate USPs and companions
have been statistically validated or confirmed, 10 for the first time,
including 7 USPs. Almost all candidates, and all validated planets, are small
(radii Rp<=3 R_E) with a median radius of R_p=1.1 R_E; the validated and
confirmed candidates have radii between 0.4 R_E and 2.4 R_E and periods from
P=0.18 to 0.96 d. The lack of candidate (a) ultra-hot-Jupiters (R_p>10 R_E) and
(b) short-period desert (3<=Rp<=10 R_E) planets suggests that both populations
are rare, although our survey may have missed some of the very deepest
transits. These results also provide strong evidence that we have not reached a
lower limit on the distribution of planetary radius values for planets at close
proximity to a star, and suggest that additional improvements in photometry
techniques would yield yet more ultra-short-period planets. The large fraction
of USPs in known multi-planet systems supports origins models that involve
dynamical interactions with exterior planets coupled to tidal decay of the USP
orbits.
