Independent transversals in r-partite graphs

Let script G sign(r, n) denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of G ∈ script G sign(r, n) is an independent set consisting of exactly one vertex from each vertex class. Let Δ(r, n) be the maximal integer such that every G ∈ script G sign(r, n) with maximal degree less than Δ(r, n) contains an independent transversal. Let C_r = lim_n→∞ Δ(r, n)/n. We establish the following upper and lower bounds on C_r, provided r > 2: 2^{[log r]-1}/2[log r] - 1 ≥ C_r ≥ max {1/2e, 1/2[log(r/3)], 1/3 . 2[log r]-3}. For all r > 3, both upper and lower bounds improve upon previously known bounds of Bollobás, Erdos and Szemerédi. In particular, we obtain that C₄ = 2/3, and that lim_r→∞ C_r ≥ 1/(2e), where the last bound is a consequence of a lemma of Alon and Spencer. This solves two open problems of Bollobás, Erdos and Szemerédi.