We present single-crystal magnetic diffuse neutron-scattering measurements with polarization analysis on the spin ice Ho_2-x Y_xTi₂O₇ (x=0, 0.3, and 1). At 2 K, the spin-flip scattering patterns are typical of a nearest-neighbor spin ice and the effects of Y doping are seen in the widths of the pinch points. At low temperature, the spin-flip patterns are characteristic of a dipolar spin ice and are unaffected by Y doping. In the nonspin-flip channel, we observe the signature of strong antiferromagnetic correlations at the same temperature as the dipolar spin-ice correlations appear in the spin-flip channel. These experimental observations have been well reproduced by our Monte Carlo simulations.