Ramsey property for finite-dimensional n-space-tuples

Determine whether the family of n-space-tuples in a countably infinite-dimensional vector space over a finite field has the Ramsey property under the substructure relation defined by coordinatewise containment, equivalently whether every finite coloring of n-space-tuples based on t-dimensional subspaces admits a monochromatic n-space-tuple of dimension k for sufficiently large ambient dimension.

Background

An n-space-tuple based on a finite subspace U is a tuple of affine-like subspaces (U+v1,...,U+vn), where v1,...,vn are linearly independent over U. The paper uses a Ramsey theorem for such tuples as the key missing ingredient for extending the positive result from distinguished subspaces of dimension at most one to arbitrary finite dimension. The case n=1 is identified with the classical Ramsey theorem for affine spaces.

References

Question D. Let V be a vector space of dimension w over a finite field. Let n E w. When U is a finite subspace of V and v1, ... , Un is a sequence linearly independent over U we call the tuple (U + v1, ... , U + Un) an n-space-tuple based on U. When (W +v, ... , W + v1) is another n-space-tuple, then by (U + v1, . .. , U + vn) < (W + vi, ... , W + vn) we denote the situation when each U + vi is a subspace of W + vị. Does the family of n-space-tuples of the space V have the Ramsey property with respect to substructure relation ≤?

— Ramsey property for spaces with bilinear forms  (2503.08312 - Ivanov et al., 11 Mar 2025) in Question D, Section 2.1