Flag algebras

Abstract

Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated as follows. For finite models $M, N$ of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let $p(M,N)$ be the probability that a randomly chosen sub-model of $N$ with $|M|$ elements is isomorphic to $M$. Which asymptotic relations exist between the quantities $p(M₁,N),…, p(M_h,N)$, where $M_1,…,M_h$ are fixed “template” models and $|N|$ grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density.