Natural statements independent from true $\Pi^0_2$ sentences

I passed this question on to Harvey Friedman, who provided the following information. Friedman has shown that the following statement is equivalent to the 2-consistency of PA:

For every recursive function $f:{\mathbb N}^k \to {\mathbb N}^k$, there exists $n_1 < \cdots < n_{k+1}$ such that $f(n_1,\ldots,n_k) \le f(n_2, \ldots, n_{k+1})$ coordinatewise.

Friedman also says that there are versions of Paris-Harrington and Kruskal's tree theorem that work. For example, "Every infinite recursive sequence of finite trees has a tree that is inf-preserving-embeddable into a later tree" is equivalent to the 2-consistency of $\Pi^1_2$ bar induction.

Friedman refers to the introduction of his forthcoming book Boolean Relation Theory and Incompleteness (downloadable from his website) for more information.