Flag Algebras Alexander A. Razborov TexPoint fonts used in EMF. Institute for Advanced Study and Steklov Mathematical Institute DIMACS Workshop on Large Graphs, October 18, 2006 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

Identifying mathematical structure behind common arguments in the area of asympotic extremal combinatorics (aka Turan densities), as well as replacing ε/δ stuff with analytic arguments is natural and highly desirable. Very closely related to the theory of graph homomorphisms by Lovász et. al (mostly independent, partially influenced). Single-purposed (so far): heavily oriented toward problems in asymptotic extremal combinatorics.

Main differencies from graph homomorphisms Our framework is specifically aimed at combining counting (Cauchy-Schwarz style) with various inductive arguments. Work with arbitrary universal first-order theories in predicative languages (digraphs, hypergraphs etc.)... (with a great deal of regret we must say farewell to W: [0,1]² → [0,1]) Work with induced objects rather than with homomorphisms (more general and intuitive in this context).

(similar to k-labelled graphs, but we insist on exact copy of σ) Definition. A type σ is a model on the ground set {1,2…,k} for some k called the size of σ Definition. A flag F of type σ is a pair (M,θ), where θ is an induced embedding of σ into M (similar to k-labelled graphs, but we insist on exact copy of σ) σ M θ 1 2 k …

F p(F1, F) – the probability that randomly chosen sub-flag of F is isomorphic to F1 F1 σ

Ground set

Multiplication F σ F1 F2

Semantics (easy part of Lovász-Szegedy)

Structure

Averaging F F1 σ F1 σ F1 σ Relative version

Cauchy-Schwarz

Upward operators (π-operators) Nature is full of such homomorphisms, and we have a very general construction (based on the logical notion of interpretation) covering most of them.

Extremal homomorphisms

Differential operators

Ensembles of random homomorphisms (or trying to salvate from Lovász-Szegedy what we need)

Applications: triangle density Partial results: Goodman [59]; Bollobás [75]; Lovász, Simonovits [83]; Fisher [89] We completely solve this for triangles (r=3)

Applications: forbidden 3-hypergraphs Not properly written down or even checked!!!

Conclusion Mathematically structured approaches (like the one presented here) is certainly no guarantee to solve your favorite extremal problem… but you are just better equipped with them. The most interesting general question: can any true inequality be proved by manipulating with finitely many finite structures (flags)? In the framework of flag algebras we have several rigorous refinements of this question.