If we would drop transitivity from intutionistic Kripke frames

Question

If we would drop transitivity from intutionistic Kripke frames, what would be an example of a propositional formulas in the implicational fragment that would not be anymore generally valid?

Answer 1

The formula

$$\varphi=(p\rightarrow q)\rightarrow(r\rightarrow(p\rightarrow q)).$$

is valid on all Kripke frames, but may fail in non-transitive frames. Consider three worlds $w_1,w_2,w_3$ such that $w_1Rw_2$ and $w_2Rw_3$ but not (!) $w_1Rw_3$, and assume that $r$ is forced at $w_2$ and $w_3$ while $p$ is forced at $w_3$ (this setting satisfies the persistency condition for atoms). Then $\varphi$ fails at $w_1$.