Induced subgraph density. VII. The five-vertex path

We prove the Erdős–Hajnal conjecture for the five-vertex path (Formula presented.); that is, there exists (Formula presented.) such that every (Formula presented.) -vertex graph with no induced (Formula presented.) has a clique or stable set of size at least (Formula presented.). This completes the verification of the Erdős–Hajnal conjecture for all five-vertex graphs. Our methods combine probabilistic and structural ideas with the iterative sparsification framework introduced in the third and fourth papers in the series.