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

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Arnosti's RANDOM-VERTEX versus RANKING conjecture for greedy matching parent of this event

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. parent of this event

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

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Original conjecture: after Theorem 2, p. 136 evidence for this event

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