Full dense-regime realization for every minimum degree

Construct, for every fixed integer d≥2 and every rational number w∈Q∩[d/2,d−1], a finite graph F with minimum degree δ(F)=d and weak saturation limit w_F=w.

Background

The paper proves that, for a prescribed minimum degree d, rational weak saturation limits in the interval [d/2,d−1] can be realized for d≤4 and establishes a substantial subinterval for d≥6. The remaining conjecture asks for the entire interval for every d≥2, including the technically difficult case d=5. The authors specifically mention two possible routes: modifying the subdivision constructions used for d=3 and d=4, or improving expansion bounds for 5-regular graphs.