Formalizing Flag Algebras in Lean

Abstract

Razborov's flag algebra method is one of the most successful tools for proving asymptotic density inequalities in extremal graph theory. It reduces an inequality about all large graphs avoiding a forbidden pattern to finitely many algebraic conditions, and a certificate satisfying them can be found by semidefinite programming. We formalize the method for finite simple graphs in Lean 4, building on Mathlib. The formalization covers flags and their densities, the flag algebra as a quotient vector space, positive homomorphisms as limits of graph sequences, Razborov's random extensions, and the downward operator that averages out labels. On top of this theory, we build verified computation of flag densities and a tactic that turns certificates found by semidefinite programming into Lean proofs. The certificate serves only as candidate data; Lean checks every fact the proof uses. With this development we verify seven Tur\'an-type upper bounds. Adding lower-bound constructions proved in Lean yields the asymptotic form of Mantel's theorem and the asymptotic Erd\H{o}s pentagon theorem $\pi(C_5;K_3)=24/625$; to our knowledge, the latter has not previously been formalized in a proof assistant. We also prove two inequalities of Goodman. All density theorems reported here depend only on the standard axioms of Lean's logic. Formalization also led us to treat forbidden subgraphs with a different semantics. Instead of building a separate flag algebra for each forbidden family, we keep a single algebra of all graphs and impose the constraint only when interpreting an inequality, through randomly labeled limits of graphs satisfying it. We prove that this ensemble semantics yields the intended density bounds, and that a condition we call root-plantability characterizes when it agrees with the standard semantics.