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, however, is not tight when $δ=1/r-ε$, that is, when the graph has edge density $1-ε$ for small $ε$. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erdős--Stone problem for all edge densities. More precisely, we prove that for every integer $r\geq 2$ and $0< δ< 1/r$ we have $$ b_n(r,δ)=Θ\left(\frac{\log n}{(1/r-δ)r\log(1/δ)}\right)\;. $$ The lower bound is obtained using a Kövari-Sós-Turán-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.