Close the logarithmic competitive-ratio gap

Close the remaining logarithmic gap between the known O(q) upper bound and the Omega(q/log q) lower bound for prophet inequalities under intersections of q unit-capacity partition matroids.

Background

The paper proves an Omega(q/log q) lower bound for prophet inequalities under intersections of q partition matroids with unit-capacity blocks, while prior work provides linear O(q) upper bounds. Thus, the optimal dependence on the number q of intersected partition matroids is determined only up to a logarithmic factor. The conclusion explicitly identifies eliminating this gap as unresolved.