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

Literature 1.L. Lovász. Large Networks and Graph Limits, American Mathematical Society, A ``canonical’’ comprehensive text on the subject. 2.A. Razborov, Flag Algebras: an Interim Report, in the volume „The Mathematics of Paul Erdos II”, Springer, 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).

Problems: Turán densities T is a universal theory in a language without constants of function symbols. Graphs, graphs without induced copies of H for a fixed H, 3-hypergraphs (possibly also with forbidden substructures), digraphs, tournaments, any relational structure. 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 N induce a sub-model isomorphic to M. What can we say about relations between p(M 1, N), p(M 2, N),…, p(M h, N) for given templates M 1,…, M h ?

Example: Mantel-Turán Theorem

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

Crash course on flag algebras

What can we say about relations between p(M 1, N), p(M 2, N),…, p(M h, N) for given templates M 1,…, M h ? What can we say about relations between φ(M 1 ), φ(M 2 ),…, φ(M h ) for given templates M 1,…, M h ?

N M

Ground set N M

N M1M1 M2M2 Models can be also multiplied

And, incidentally, where are our flags? NSF

Definition. A type σ is 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 σ is a partially labeled model, i.e. a pair (M,θ), where θ is an induced embedding of the type σ into M.

F Averaging (= label erasing) F1F1 σ F1F1 σ F1F1 σ

Plain methods (Cauchy-Shwarz):

Notation (in the asymptotic form)

Clique densit y Partial results on computing g r (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.

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

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

Tetrahedron Problem

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.

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.

Drawback: relevant only to Turán’s original example.

Cacceta-Haggkvist conjecture

Erdös’s Pentagon Problem [Hladký Král H. Hatami Norin 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 C 5, cycles of length 5.

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

2/3 conjecture [Erdös Faudree Gyárfás Schelp 89]

Pure inducibility Ordinary graphs

Oriented graphs

Minimum inducibility (for tournaments)

3-graphs

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.

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?

Thank you