Partition relations on a plain product order type

The goal of this short note is to interest set theorists in the order type ω ∗ ⋅ ω 1 , and to encourage them to work on the question of whether or not the Continuum Hypothesis decides the partition relation τ → ( τ , infinite path ) 2 , for τ = ω ∗ ⋅ ω 1 and for τ = ω 1 ω + 2 .