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 ( M 1 , N ),…, p ( M h , N ), where M 1 ,…, M 1 , 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.