Iterated forcing in quadratic form theory

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has its orthogonal complement of at most countable dimension is independent of the axioms of ZFC set theory. Such a space will be called astrong Gross space in the sequel. Cardinal invariants of the continuum decide whether strong Gross spaces exist or not. Namely, when b=ω₁ a strong Gross space of dimension ℵ₁ exists. When p>ω₁ such spaces do not exist. Here we answer the question what happens with strong Gross spaces in case b>ω₁ or p=ω₁. This work forms part of the author’s Habilitationsschrift at the ETH Zürich. The author is supported by the Basic Research Foundation of the Israel Academy of Sciences.