A Tight Erdős-Stone Bound for All Graph Densities

The Erdős--Stone Theorem asserts that if a graph has edge density $1-1/r+δ$ then it contains a complete $(r+1)$-partite graph with $b$ vertices in each part, where $b=b_n(r,δ) \gg 1$. The celebrated Chvátal--Szemerédi theorem determined the exact order of $b_n(r,δ)$ for every $δ< 1/r^3$. Their bound