An average degree condition for independent transversals

In 1994, Erd?s, Gyárfás and ?uczak posed the following problem: given disjoint vertex sets V1,…,Vn of size k, with exactly one edge between any pair Vi,Vj, how large can n be such that there will always an independent transversal? They showed maximal is at most (1+o(1))k2, by providing explicit construction these parameters no transversal. also proved a lower bound which smaller 2e-factor. this paper, we solve problem showing their upper best possible: if n?(1?o(1))k2, fact, result very special case much more general theorem concerns transversals in arbitrary partite graphs are ‘locally sparse’, meaning maximum degree each parts relatively small. setting, Loh Sudakov provided global condition for existence We show relaxed to average condition. use our new establish tight bounds version Erd?s–Gyárfás–?uczak conjecture Yuster from 1997. This exploits connection Turán numbers complete bipartite graphs, might interest.