Independent transversals in r-partite graphs

Let G(r,n) denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of G is an element of G(r,n) is an independent set consisting of exactly one vertex from each vertex class. Let Delta(r,n) be the maximal integer such that every G is an element of G(r,n) with maximal degree less than Delta(r,n) contains an independent transversal. Let C-r = lim(n-->infinity)Delta(r,n)/n. We establish the following upper and lower bounds on C-r, provided r > 2: 2(right perpendicular log rleft perpendicular?(-1))/2(right perpendicular rleft perpendicular)-1 greater than or equal to C-r greater than or equal to max {1/2e,1/2(inverted right perpendicular log(r/3)inverted left perpendicular), 1/3.2(inverted right perpendicular rinverted left perpendicular-3)}. For all r > 3, both upper and lower bounds improve upon previously known bounds of Bollobas, Erdos and Szemeredi. In particular, we obtain that C-4 = 2/3, and that lim(r-->infinity) C-r greater than or equal to 1/(2e), where the last bound is a consequence of a lemma of Alon and Spencer. This solves two open problems of Bollobas, Erdos and Szemeredi.