Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph K_n has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost n. In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured K_n has a rainbow cycle of length n − O(n^3/4). One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured K_n formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular, it has very good expansion properties.