Spaces of symmetric matrices containing a nonzero matrix of bounded rank (Q1301315)

Let \(S_n(F)\) be the space of all \(n\times n\) 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 \(S_n(F)\) contains a nonzero matrix whose rank is at most \(k\). For \(F=\mathbb{C}\), it is known that \(d(n,k,\mathbb{C})={n-k+1\choose 2}+1\). But to evaluate \(d(n,k,\mathbb{R})\) is more difficult. In this paper, using a simple example and some classical results, it is computed that \(d(n,1,\mathbb{R})={n+1\choose 2}\neq d(n,1,\mathbb{C})\), and \[ d(n,n-1,\mathbb{R})=\begin{cases} 2&\text{if \(n\) is odd,}\\ \rho(n/2)+2&\text{otherwise,}\end{cases} \] where \(\rho(n)\) is the Radon-Hurwitz number. As the main object of this paper, some partial results regarding \(d(n,n-2,\mathbb{R})\) are obtained, in particular it is shown that \(4\leq d(4,2,\mathbb{R})\leq 5\). Furthermore, in a joint paper of the same authors and \textit{D. Falikman} which is in preparation, it is shown that \(d(4,2,\mathbb{R})=5\).