Infinite image partition regular matrices

Determine whether the matrix-version conclusion of the Moreira theorem remains true when A and B are infinite image partition regular matrices; that is, establish whether every finite coloring of the natural numbers admits vectors X and Y for which the sets of entries of AX, AX + BY, and AX . BY are monochromatic, with addition and multiplication defined coordinate-wise.

Background

The paper proves that if A and B are finite image partition regular matrices of the same order, then every finite coloring of the natural numbers contains vectors X and Y such that the entries of AX, AX + BY, and AX . BY are monochromatic. This extends Moreira’s result for the scalar case A = B = (1), which yields a monochromatic configuration of the form {x, x + y, xy}.

The authors explicitly note that their proof does not establish the analogous result for infinite image partition regular matrices. Thus, it remains unresolved whether the finite-matrix theorem extends to the infinite setting.