  Cold accretion is a primary growth mechanism of simulated galaxies, yet
observational evidence of "cold flows" at redshifts where they should be most
efficient ($z=2$-4) is scarce. In simulations, cold streams manifest as
Lyman-limit absorption systems (LLSs) with low heavy-element abundances similar
to those of the diffuse IGM. Here we report on an abundance survey of 17 H
I-selected LLSs at $z=3.2$-4.4 which exhibit no metal absorption in SDSS
spectra. Using medium-resolution spectra obtained at Magellan, we derive
ionization-corrected metallicities (or limits) with a Markov-Chain Monte Carlo
sampling that accounts for the large uncertainty in $N_{\rm HI}$ measurements
typical of LLSs. The metal-poor LLS sample overlaps with the IGM in metallicity
and is best described by a model where $71^{+13}_{-11}\%$ are drawn from the
IGM chemical abundance distribution. These represent roughly half of all LLSs
at these redshifts, suggesting that 28-40$\%$ of the general LLS population at
$z\sim3.7$ could trace unprocessed gas. An ancillary sample of ten LLSs without
any a priori metal-line selection is best fit with $48^{+14}_{-12}\%$ of
metallicities drawn from the IGM. We compare these results with regions of a
moving-mesh simulation; the simulation finds only half as many baryons in
IGM-metallicity LLSs, and most of these lie beyond the virial radius of the
nearest galaxy halo. A statistically significant fraction of all LLSs have low
metallicity and therefore represent candidates for accreting gas; large-volume
simulations can establish what fraction of these candidates actually lie near
galaxies and the observational prospects for detecting the presumed hosts in
emission.
