The Erdős–Hajnal Conjecture states that there is a constant $c(H) > 0$ for each $H$ such that we can take $f(n) = n^{c(H)}$ in the above formulation.

Nguyen, Scott, and Seymour [NSS23] proved the conjecture for $H = P_5$, the path on five vertices: every $P_5$-free graph on $n$ vertices has a clique or independent set of polynomial size.

[NSS23] Nguyen, T., Scott, A. and Seymour, P., Induced subgraph density. VII. The five-vertex path. arXiv:2312.15333