Local flag algebras

2026-07-14

AI Summary

This work addresses extremal graph theory problems where bounds depend on the maximum degree Δ(G) rather than the order |G| of the graph. It introduces a novel framework termed “local flag algebra,” which innovatively replaces the classical density normalization based on |G| with one based on Δ(G), while preserving the semidefinite programming machinery of traditional flag algebras. This extension significantly broadens the applicability of flag algebra methods under maximum-degree constraints. As a key application, the framework yields the first tight upper bound on the number of pentagons in triangle-free graphs that explicitly depends on both |G| and Δ(G), demonstrating its effectiveness in capturing finer structural dependencies.

Abstract

We introduce local flag algebras, a variant of Razborov's flag algebra framework in which densities are normalised by the maximum degree $Δ(G)$ rather than the order $|G|$. 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 $|G|$ and $Δ(G)$.

Problem

• local flag algebras
• extremal problems
• maximum degree
• triangle-free graph
• pentagons

Innovation

• local flag algebras
• maximum degree
• extremal graph theory
• semidefinite programming
• triangle-free graphs