Spaces of symmetric matrices containing a nonzero matrix of bounded rank

AbstractLet Sn(F) denote the space of all n × n symmetric matrices over the field F. Given a positive integer k such that k < n, let d(n, k, F) be the smallest integer ℓ such that every ℓ dimensional subspace of Sn(F) contains a nonzero matrix whose rank is at most k. It is our purpose to consider d(n, k, F) for ℝ and ℂ. While the computation of d(n, k, Fℂ) is quite straightforward, we point out the difficulty in evaluating d(n, k, Fℝ). We obtain partial results regarding d(n, nℒℝ), and in particular show that 4⩽d(4,2,ℝ)