Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity

We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let ω(G) denote the largest size of a pairwise noncommuting subset of a group G, let a(G) be the least size of an abelian cover, and define h(n) = sup{a(G) : ω(G) ≤ n}. Answering a quantitative question posed by Erdős, we prove log₂ h(n) = n/2 + o(n), equivalently h(n)^(1/n) → √2. Thus the exact exponential rate is √2, sharpening the previously known general exponential upper and lower bounds. Extraspecial 2-groups provide the matching lower bound. The upper bound combines a central-factor analysis of finite p-groups with alternating-form clique estimates, interaction control across central factors, Sylow decomposition, and a polynomial-cost reduction to the Fitting subgroup. As consequences, we determine the same sharp exponential rate for the minimum possible index of an abelian subgroup and show that asymptotic extremality is concentrated in 2-groups. In graph-theoretic terms, the result determines the sharp asymptotic chromatic-versus-clique growth rate for noncommuting graphs of groups. This result resolves Erdős Problem #117 at the level of its sharp exponential asymptotics.