On the ${\mathrm{\Pi }}_2^1$ consequences of ${\mathrm{\Pi }}_1^1$-${\mathsf{C}\mathsf{A}}_0$

In this paper, we introduce a hierarchy dividing the set $\{\sigma \in {\mathrm{\Pi }}_2^1:{\mathrm{\Pi }}_1^1$-${\mathsf{C}\mathsf{A}}_0⊢\sigma \}$. Then, we give some characterizations of this set using weaker variants of some principles equivalent to ${\mathrm{\Pi }}_1^1$-${\mathsf{C}\mathsf{A}}_0$: leftmost path principle, Ramsey’s theorem for ${\mathrm{\Sigma }}_n^0$ classes of $[\mathbb{N}{]}^{\mathbb{N}}$ and determinacy for $({\mathrm{\Sigma }}_1^0{)}_n$ classes of ${\mathbb{N}}^{\mathbb{N}}$.