Spaces of symmetric matrices containing a nonzero matrix of bounded rank

Let S-n(F) denote the space of all n x n symmetric matrices over the field F. Given a positive integer k such that k < n, let d(n, k, F) be the smalle st integer l such that every l dimensional subspace of S-n(F) contains a no nzero matrix whose rank is at most k. It is our purpose to consider d(n,k,F ) for F = R and F = C. While the computation of d(n, k, C) is quite straigh tforward, we point out the difficulty in evaluating d(n, k, R). We obtain p artial results regarding d(n, n-2, R), and in particular show that 4 less t han or equal to d(4, 2, R) less than or equal to 5. (C) 1999 Elsevier Scien ce Inc. All rights reserved.