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In 1994, Erdős, Gyárfás and Łuczak posed the following problem: given disjoint vertex sets V 1 , ... , V n of size k , with exactly one edge between any pair V i , V j , how large can n be such that there will always be an independent transversal? They showed that the maximal n is at most (1 + o (1)) k 2 , by providing an explicit construction with these parameters and no independent transversal. They also proved a lower bound which is smaller by a 2 e -factor. In this paper, we solve this problem by showing that their upper bound construction is best possible: if n ≤ (1 − o (1)) k 2 , there will always be an independent transversal. In fact, this result is a very special case of a much more general theorem which concerns independent transversals in arbitrary partite graphs that are 'locally sparse', meaning that the maximum degree between each pair of parts is relatively small. In this setting, Loh and Sudakov provided a global maximum degree condition for the existence of an independent transversal. We show that this can be relaxed to an average degree condition. We can also use our new theorem to establish tight bounds for a more general version of the Erdős–Gyárfás–Łuczak problem and solve a conjecture of Yuster from 1997. This exploits a connection to the Turán numbers of complete bipartite graphs, which might be of independent interest. [ABSTRACT FROM AUTHOR]

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. Our main result implies that, given any optimal colouring of a sufficiently large complete graph K 2 n , there exists a decomposition of K 2 n into isomorphic rainbow spanning trees. This settles conjectures of Brualdi–Hollingsworth (from 1996) and Constantine (from 2002) for large graphs. [ABSTRACT FROM AUTHOR]