Spaces of symmetric matrices of bounded rank (Q1319988)

This paper concerns certain subspaces of the space \({\mathcal S}_ n(F)\) of all \(n\times n\) symmetric matrices over a field \(F\). Given \(t < n\), a subspace \(W\) of \({\mathcal S}_ n (F)\) is a \(\overline t\)-subspace if it consists only of matrices of rank at most \(t\). It is shown that a \(\overline t\)-subspace of maximal possible dimension is congruent to one of two possible types of subspace. The argument is of a graph-theoretical nature.