The value of f(n,2) is the subject of a conjecture by Kézdy and Snevily that implies two famous conjectures by Ryser and Brualdi on transversals in Latin squares. We show that f(n,2) ≤ n+O(log n) for all n and that f(n,2) ≤ n+2 whenever n=3m for m > 1. We also construct, for each odd m ≥ 3, a Latin square of order 3m with two rows that each contain 2m-2 transversal-free entries. This gives the best upper bound yet on the number of disjoint transversals for an infinite family of Latin squares of odd order.