1. 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.
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2. 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.

3. 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).

4. (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 …

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

6. Ground set

7. Multiplication F σ F1 F2

8. Semantics
(easy part of Lovász-Szegedy)

9. Structure

10. Averaging F F1 σ F1 σ F1 σ
Relative version

11. Cauchy-Schwarz

12. 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.

14. Extremal homomorphisms

15. Differential operators

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

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

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

22. 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.