  I develop in depth the machinery of $(\mathcal L, n)$-models originally
introduced by Shelah and, independently in a slightly different form by Kripke.
This machinery allows fairly routine constructions of true but unprovable
sentences in $\mathsf{PA}$. I give two applications: 1. Shelah's alternative
proof of the Paris-Harrington theorem, and 2. The independence over
$\mathsf{PA}$ of a new $\Pi^0_1$ Ramsey theoretic statement about colorings of
finite sequences of structures.
