Packing paths in complete graphs

Let λKn denote the complete graph of order n and multiplicity λ. We prove Tarsi's conjecture [M. Tarsi, Decomposition of a complete multigraph into simple paths: Nonbalanced handcuffed designs, J. Combin. Theory Ser. A 34 (1983) 60–70] that for any positive integers n, λ and t, and any sequence m1,m2,…,mt of positive integers, it is possible to pack t pairwise edge-disjoint paths of lengths m1,m2,…,mt in λKn if and only if mi⩽n−1 for i=1,2,…,t and .