  We present new, deep, narrow- and broad-band Hubble Space Telescope
observations of seven of the most star-forming brightest cluster galaxies
(BCGs). Continuum-subtracted [O II] maps reveal the detailed, complex structure
of warm ($T \sim 10^4$ K) ionized gas filaments in these BCGs, allowing us to
measure spatially-resolved star formation rates (SFRs) of ~60-600 Msun/yr. We
compare the SFRs in these systems and others from the literature to their
intracluster medium (ICM) cooling rates (dM/dt), measured from archival Chandra
X-ray data, finding a best-fit relation of log(SFR) = (1.67+/-0.17) log(dM/dt)
+ (-3.25+/-0.38) with an intrinsic scatter of 0.39+/-0.09 dex. This
steeper-than-unity slope implies an increasingly efficient conversion of hot
($T \sim 10^7$ K) gas into young stars with increasing dM/dt, or conversely a
gradual decrease in the effectiveness of AGN feedback in the strongest cool
cores. We also seek to understand the physical extent of these multiphase
filaments that we observe in cluster cores. We show, for the first time, that
the average extent of the multiphase gas is always smaller than the radii at
which the cooling time reaches 1 Gyr, the tcool/tff profile flattens, and that
X-ray cavities are observed. This implies a close connection between the
multiphase filaments, the thermodynamics of the cooling core, and the dynamics
of X-ray bubbles. Interestingly, we find a one-to-one correlation between the
average extent of cool multiphase filaments and the radius at which the cooling
time reaches 0.5 Gyr, which may be indicative of a universal condensation
timescale in cluster cores.
