Ramsey property relative to a prescribed subspace in a bilinear space with boundedly many hyperbolic pairs

Determine whether, for a vector space over GF(2) equipped with a skew-symmetric bilinear form having a finite positive number of hyperbolic pairs and infinite-dimensional radical, the age satisfies the Ramsey property with respect to copies of every finite subspace A whose quotient dimension over its intersection with the radical is at least one.

Background

Section 2 studies a space over GF(2) decomposed into finitely many hyperbolic subspaces together with an infinite-dimensional radical. The authors establish failure of the Ramsey property for broad classes of pairs of finite subspaces and obtain positive results under additional structural restrictions. Question B asks for a general classification relative to a fixed finite subspace A that has a nontrivial component outside the radical.