The stopping redundancy of a linear code is defined as the minimum number of rows in its parity-check matrix such that the smallest stopping sets have size equal to the minimum distance of the code. We derive new upper bounds on the stopping redundancy of maximum distance separable (MDS) codes, and show how they improve upon previously known results. The new bounds are found by upper bounding the stopping redundancy by a combinatorial quantity closely related to Turan numbers. (The Turan number, T(v, k, t), is the smallest number of t-subsets of a v-set, such that every k-subset of the v-set contains at least one of the t-subsets.) Asymptotically, we show that the stopping redundancy of MDS codes with length n and minimum distance d > 1 is T(n, d -1, d - 2)(1 + O(n-1)) for fixed d, and is at most T(n, d - 1, d - 2)(3 + O(n-1)) for fixed code dimension k = n - d + 1. For d = 2,3,4, we prove that the stopping redundancy is equal to T(n, d - 1, d - 2). For d = 5, we show that the stopping redundancy is either T(n, 4, 3) or T(n, 4, 3) + 1