Owings’s original problem on infinite monochromatic sumsets in the natural numbers (2 colours)

Determine whether the natural numbers N satisfy N → (ℵ0)2+·; equivalently, decide whether every 2-colouring c: N → 2 admits an infinite subset X ⊆ N such that the sumset X + X is monochromatic (i.e., all elements of X + X have the same colour).

Background

The paper studies the Ramsey-theoretic statement G → (κ)θ+·, which asks that every colouring c: G → θ yields a subset X ⊆ G of size κ with X + X monochromatic. Owings’s 1974 problem is the case G = N, κ = ℵ0, and θ = 2.

It is known that the analogous statement fails for θ = 3 (there exists a 3-colouring with no infinite monochromatic X + X), and even with additional density constraints, but the 2-colour case remains unresolved.