  We present the small-scale (0.01<r<8 h^{-1} Mpc) projected correlation
function w_p(r_p) and real space correlation function xi(r) of 24520 luminous
early-type galaxies from the Sloan Digital Sky Survey Luminous Red Galaxy (LRG)
sample (0.16<z<0.36). ``Fiber collision'' incompleteness of the SDSS
spectroscopic sample at scales smaller than 55 arcseconds prevents measurements
of the correlation function for LRGs on scales smaller than ~0.3 Mpc by the
usual methods. In this work, we cross-correlate the spectroscopic sample with
the imaging sample, with a weighting scheme to account for the collisions,
extensively tested against mock catalogs. We correct for photometric biases in
the SDSS imaging of close galaxy pairs. We find that the correlation function
xi(r) is surprisingly close to a r^{-2} power law over more than 4 orders of
magnitude in separation r. This result is too steep at small scales to be
explained in current versions of the halo model for galaxy clustering. We infer
an LRG-LRG merger rate of less than or similar to 0.6 Gyr^{-1} Gpc^{-3} for
this sample. This result suggests that the LRG-LRG mergers are not the main
mode of mass growth for LRGs at z<0.36.
