Arnosti's RANDOM-VERTEX versus RANKING conjecture for greedy matching

For every finite instance in the stated model, every common capacity $C\ge1$ and every threshold $K$, we prove $$\Pr(M_{\mathrm{RV}}\ge K)\ge\Pr(M_{\mathrm{RANKING}}\ge K).$$ The comparison holds for fixed arrival and bin-priority orders independent of the neighborhoods; averaging gives Arnosti's random-order model, including the conjectured $C>2$ regime. The proof combines capped-load association, a two-bin coefficient identity, a saturation bound, refined histories and coupling at accepted-job counts. The written proof uses Cohen–Sackrowitz (1987), Lemma 4.1; Lean proves the required finite rational association statement directly. The inequality compares matching-size distributions. It does not assert dominance on every run, conditional on an arbitrary fixed graph, or for unequal capacities.

For every finite instance in the stated model, every common capacity $C\ge1$ and every threshold $K$, we prove $$\Pr(M_{\mathrm{RV}}\ge K)\ge\Pr(M_{\mathrm{RANKING}}\ge K).$$ The comparison holds for fixed arrival and bin-priority orders independent of the neighborhoods; averaging gives Arnosti's random-order model, including the conjectured $C>2$ regime. The proof combines capped-load association, a two-bin coefficient identity, a saturation bound, refined histories and coupling at accepted-job counts. The written proof uses Cohen–Sackrowitz (1987), Lemma 4.1; Lean proves the required finite rational association statement directly. The inequality compares matching-size distributions. It does not assert dominance on every run, conditional on an arbitrary fixed graph, or for unequal capacities.