Perfect fractional matchings in random hypergraphs

Given an r-uniform hypergraph H = (V, E) on \V\=n vertices, a real-valued function f : -E→R⁺ is called a perfect fractional matching if ∑ _v∈e f(e) ≤ 1 for all v ∈ V and ∑ _e∈E f(e) = n/r. Considering a random r-uniform hypergraph process of n vertices, we show that with probability tending to 1 as n→∞, at the very moment t₀ when the last isolated vertex disappears, the hypergraph H,_to has a perfect fractional matching. This result is clearly best possible. As a consequence, we derive that if p(n) = (In n + w(n))/ (n-1/r-1), where w(n) is any function tending to infinity with n, then with probability tending to 1 a random r-uniform hypergraph on n vertices with edge probability p has a perfect fractional matching. Similar results hold also for random r-partite hypergraphs.