Understanding the evolution of carbon and iron in the Milky Ways halo is of importance because these two elements play crucial roles constraining star formation, Galactic assembly, and chemical evolution in the early Universe. Here, we explore the spatial distributions of carbonicity, [C/Fe], and metallicity, [Fe/H], of the halo system based on medium-resolution ($R sim$ 1,300) spectroscopy of $sim$58,000 stars in the Southern Hemisphere from the AAOmega Evolution of Galactic Structure (AEGIS) survey. The AEGIS carbonicity map exhibits a positive gradient with distance, as similarly found for the Sloan Digital Sky Survey (SDSS) carbonicity map of Lee et al. The metallicity map confirms that [Fe/H] decreases with distance, from the inner halo to the outer halo. We also explore the formation and chemical-evolution history of the halo by considering the populations of carbon-enhanced metal-poor (CEMP) stars present in the AEGIS sample. The cumulative and differential frequencies of CEMP-no stars (as classified by their characteristically lower levels of absolute carbon abundance, $A$(C) $leq$ 7.1 for sub-giants and giants) increases with decreasing metallicity, and is textit{substantially higher than previous determinations} for CEMP stars as a whole. In contrast, that of CEMP-$s$ stars (with higher $A$(C)), remains almost flat, at a value $sim$10%, in the range $-,4.0 lesssim$ [Fe/H] $lesssim-$2.0. The distinctly different behaviors of the CEMP-no and CEMP-$s$ stars relieve the tension with population-synthesis models assuming a binary mass-transfer origin, which previously struggled to account for the higher reported frequencies of CEMP stars, taken as a whole, at low metallicity.
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