Alexander A. Razborov University of Chicago Steklov Mathematical Institute

1. Flag Algebras: an Interim Report Alexander A. Razborov University of Chicago Steklov Mathematical Institute Toyota Technological Institute at Chicago Institute for Mathematics and Applications, September 11, 2014 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Literature L. Lovász. Large Networks and Graph Limits, American Mathematical Society, 2012. A ``canonical’’ comprehensive text on the subject. A. Razborov, Flag Algebras: an Interim Report, in the volume „The Mathematics of Paul Erdos II”, Springer, 2013. A registry of concrete results obtained with the help of the method. 3. A. Razborov, What is a Flag Algebra, in Notices of the AMS (October 2013). A high-level overview (for “pure” mathematicians).

3. Problems: Turán densities Graphs, graphs without induced copies of H for a fixed H, 3-hypergraphs (possibly also with forbidden substructures), digraphs, tournaments, any relational structure. T is a universal theory in a language without constants of function symbols. M,N two models: M is viewed as a fixed template, whereas the size of N grows to infinity. p(M,N) is the probability (aka density) that |M| randomly chosen vertices in Ninduce a sub-model isomorphic to M. What can we say about relations between p(M1, N), p(M2, N),…, p(Mh, N) for given templates M1,…, Mh?

4. Example: Mantel-Turán Theorem

5. Deviations • More complicated scenarios: • Cacceta-Haggkvist conjecture (minimum degrees) • Erdössparse halves problem (additional • structure) Beyond Turán densities: results are few and far between. [Baber 11; Balogh, Hu, Lidicḱy, Liu 12]: flag-algebraic (sort of) analysis on the hypercube Qn

6. Crash course on flag algebras

7. What can we say about relations between φ(M1), φ(M2),…, φ(Mh)for given templates M1,…, Mh? What can we say about relations between p(M1, N), p(M2, N),…, p(Mh, N) for given templates M1,…, Mh?

8. N M

9. N M Ground set

10. N M1 M2 Models can be also multiplied

12. And, incidentally, where are our flags? NSF

13. Definition. A type σis a totally labeled model, i.e. a model with the ground set {1,2…,k} for some k called the size of σ. Definition. A flag F of type σis a partially labeled model, i.e. a pair(M,θ), whereθis an induced embedding of the type σintoM.

14. F F1 F1 F1 σ σ σ Averaging (= label erasing)

15. Plain methods (Cauchy-Shwarz):

16. Notation (in the asymptotic form)

17. Clique density Partial results on computing gr (x): Goodman [59]; Bollobás [75]; Lovász, Simonovits [83]; Fisher [89] Flag algebras completely solve this for triangles (r=3). • Methods are not plain. • Ensembles of random homomorphisms (infinite • analogue of the uniform distribution over vertices, • edges etc.). Done without semantics! • Variational principles: if you remove a vertex or an edge in an extremal solution, the goal function may only increase.

19. Upper bound See [Reiher 11] for further comments on the interplay between flag algebras and Lagrangians.

20. [Das, Huang, Ma, Naves, Sudakov 12]: l=3, r=4 orl=4, r=3. More cases: l=5, r=3 andl=6, r=3 verified by Vaughan. [Pikhurko 12]: l=3, 5 ≤ r≤7.

21. Tetrahedron Problem

23. Extremal examples (after [Brown 83; Kostochka 82; Fon-der-Flaass 88]) A triple is included iff it contains an isolated vertex or a vertex of out-degree 2.

25. Some proof features. • extensive human-computer interaction. • extensively moving around auxiliary results about different theories: 3-graphs, • non-oriented graphs, oriented graphs and their • vertex-colored versions.

26. Drawback: relevant only to Turán’s original example.

28. Cacceta-Haggkvist conjecture

30. Erdös’s Pentagon Problem [HladkýKrál H. HatamiNorin R 11; Grzesik 11] [Erdös 84]: triangle-free graphs need not be bipartite. But how exactly far from being bipartite can they be? One measure proposed by Erdös: the number of C5, cycles of length 5.

31. An earlier example: clique densities. Inherently analytical and algebraic methods lead to exact results in extremalcombinatorics about finite objects.

32. 2/3 conjecture [ErdösFaudreeGyárfás Schelp 89]

33. Pure inducibility Ordinary graphs

35. Oriented graphs

36. Minimuminducibility (for tournaments)

37. 3-graphs

38. Permutations (and permutons) In our language, it is simply the theory of two linear orderings on the same ground set and, as such, does not need any special treatment. In fact, this is roughly the only other theory for which semantics looks as nice as for graphons.

40. 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. More connections to graph limits and other things?

41. Thank you