Erdős Problem 172

Reference: erdosproblems.com/172

theorem Erdos172.erdos_172 : (∀ (n : ℕ) (color : ℕ → Fin n) (m : ℕ), ∃ (A : Finset ℕ), A.card ≥ m ∧ ∃ (c : Fin n), ∀ (S : Finset { x : ℕ // x ∈ A }), S.Nonempty → color (∑ x ∈ S, ↑x) = c ∧ color (∏ x ∈ S, ↑x) = c) ↔ sorry

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?