Packing paths in complete graphs

Let λ K n denote the complete graph of order n and multiplicity λ. We prove Tarsiʹs 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 m 1 , m 2 , … , m t of positive integers, it is possible to pack t pairwise edge-disjoint paths of lengths m 1 , m 2 , … , m t in λ K n if and only if m i ⩽ n − 1 for i = 1 , 2 , … , t and m 1 + m 2 + ⋯ + m t ⩽ λ n ( n − 1 ) 2 .