Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \subseteq N$ so that $c \upharpoonright [H]^e$ is constant, and the cardinality of $H$ is larger than both $k$ and ${\rm min}(H)$. This last clause separates $\mathsf{PH}$ from finite Ramsey's theorem. Famously Paris and Harrington showed that $\mathsf{PH}$ is not provable in $\mathsf{PA}$ but it is true in the standard model. Of course in general the interest from a logical point of view is the non-provability result but in this question I want to focus on the fact that it's true in the standard model.

The standard proof of this fact mimics the proof of finite Ramsey's theorem as an application of Konig's Lemma and infinite Ramsey theorem noting that, in the infinite homogeneous set one gets, we can choose $H$ as big a finite segment as we like. If people are unclear what I mean, I can write this down. The same proof actually shows that for $any$ $f:\omega \to \omega$ we can actually arrange, in the definition of $\mathsf{PH}$ that not only $|H| > {\rm min}(H)$ but actually $|H| > f({\rm min}(H))$. Let us denote this (seemingly stronger) statement by $\mathsf{PH}_f$. So, for example, $\mathsf{PH}$ is $\mathsf{PH}_{identity}$. My question is:

When is it the case that for $f, g:\omega \to \omega$ do we have $\mathsf{PH}_f$ implies $\mathsf{PH}_g$ over $\mathsf{PA}$ as a base theory?

For example, one particular instance of this question that I'm interested in is:

Does $\mathsf{PH}$ imply $\mathsf{PH}_f$ for all computable $f$? What about all arithmetic $f$?