Local flag algebras

We introduce local flag algebras, a variant of Razborov’s flag algebra framework in which densities are normalised by the maximum degree $\mathrm{\Delta }(G)$ rather than the order $\mathrm{\mid }G\mathrm{\mid }$. The framework supports the same semidefinite-method machinery as the classical version, but is tailored to extremal problems that scale with the maximum degree. As an illustrative first application we bound the number of pentagons in a triangle-free graph $G$ as a function of $\mathrm{\mid }G\mathrm{\mid }$ and $\mathrm{\Delta }(G)$.