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• W2972456824 abstract "For fixed r≥3 and n divisible by r, let H=Hn,Mr be the random M-edge r-graph on V={1,…,n}; that is, H is chosen uniformly from the M-subsets of K:=(Vr) (:={r-subsets ofV}). Shamir's Problem (circa 1980) asks, roughly, for what M=M(n) is H likely to contain a perfect matching (that is, n/r disjoint r-sets)? In 2008 Johansson, Vu and the author showed that this is true for M>Crnlog⁡n. The present paper has two purposes. First, it establishes the asymptotically correct version of the 2008 result: Theorem 1For fixed ε>0 and M>(1+ε)(n/r)log⁡n,P(Hcontains a perfect matching)→1asn→∞. Theorem 1 For fixed ε>0 and M>(1+ε)(n/r)log⁡n,P(Hcontains a perfect matching)→1asn→∞. Second, it begins a proof of the definitive “hitting time” statement: Theorem 2If A1,… is a uniform permutation of K, Ht={A1,…,At}, andT=min⁡{t:A1∪⋯∪At=V}, then P(HTcontains a perfect matching)→1asn→∞. Theorem 2 If A1,… is a uniform permutation of K, Ht={A1,…,At}, andT=min⁡{t:A1∪⋯∪At=V}, then P(HTcontains a perfect matching)→1asn→∞. It is shown here that Theorem 2 follows from a conditional version of Theorem 1 that will be proved elsewhere. The key ideas in that proof are similar to those for Theorem 1, but the argument is a longer story, and it has seemed best to give the present separate proof of Theorem 1, in which those ideas may appear more clearly." @default.
• W2972456824 created "2019-09-19" @default.
• W2972456824 creator A5074687160 @default.
• W2972456824 date "2023-06-01" @default.
• W2972456824 modified "2023-10-14" @default.
• W2972456824 title "Asymptotics for Shamir's problem" @default.
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• W2972456824 doi "https://doi.org/10.1016/j.aim.2023.109019" @default.
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