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We present a method for selecting $z>4$ dusty, star forming galaxies (DSFGs)using Herschel/SPIRE 250/350/500 $\mu m$ flux densities to search for redsources. We apply this method to 21 deg$^2$ of data from the HerMES survey toproduce a catalog of 38 high-$z$ candidates. Follow-up of the first 5 of thesesources confirms that this method is efficient at selecting high-$z$ DSFGs,with 4/5 at $z=4.3$ to $6.3$ (and the remaining source at $z=3.4$), and thatthey are some of the most luminous dusty sources known. Comparison withprevious DSFG samples, mostly selected at longer wavelengths (e.g., 850 $\mum$) and in single-band surveys, shows that our method is much more efficient atselecting high-$z$ DSFGs, in the sense that a much larger fraction are at$z>3$. Correcting for the selection completeness and purity, we find that thenumber of bright ($S_{500\,\mu m} \ge 30$ mJy), red Herschel sources is $3.3\pm 0.8$ deg$^{-2}$. This is much higher than the number predicted by currentmodels, suggesting that the DSFG population extends to higher redshifts thanpreviously believed. If the shape of the luminosity function for high-$z$ DSFGsis similar to that at $z\sim2$, rest-frame UV based studies may be missing asignificant component of the star formation density at $z=4$ to $6$, even aftercorrection for extinction.

HerMES: Candidate High-Redshift Galaxies Discovered with Herschel/SPIRE