Alexander A. Razborov University of - PowerPoint Presentation

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Alexander A. RazborovUniversity of ChicagoSteklov Mathematical InstituteToyota Technological Institute at ChicagoInstitute for Mathematics and Applications, September 11, 2014

Flag Algebras: an Interim Report

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LiteratureL. 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).Slide3

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.

W

hat

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

?Slide4

Example: Mantel-Turán TheoremSlide5

DeviationsMore 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 hypercubeQn Slide6

Crash course on flag algebrasSlide7

What can we say about relations between p(M1, N), p

(M2, N),…,

p

(

M

h

,

N

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

φ

(

M

h

)

for given templates

M

1

,…,

M

h

?Slide8

N

MSlide9

Ground set

N

MSlide10

N

M

1

M

2

Models can be also multipliedSlide11
Slide12

And, incidentally, where are our flags?

NSFSlide13

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.Slide14

F

Averaging (= label erasing)

F

1

σ

F

1

σ

F

1

σSlide15

Plain methods (Cauchy-Shwarz):Slide16

Notation (in the asymptotic form)Slide17

Clique density

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.Slide18
Slide19

Upper bound

See

[

Reiher

11]

for further comments on the interplay

b

etween flag algebras and

Lagrangians.Slide20

[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. Slide21

Tetrahedron ProblemSlide22
Slide23

Extremal examples (after [Brown 83; Kostochka 82; Fon-der-Flaass

88])

A triple is included

iff

it contains an isolated vertex

o

r a vertex of out-degree 2. Slide24
Slide25

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.Slide26

Drawback: relevant only to

Turán’s original example.Slide27
Slide28

Cacceta-Haggkvist conjectureSlide29
Slide30

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 C5, cycles of length 5. Slide31

Inherently analytical and algebraic methodslead to exact results in extremal combinatorics about finite objects.An earlier example: clique densities.Slide32

2/3 conjecture [Erdös Faudree Gyárfás

Schelp 89]Slide33

Pure inducibility

Ordinary graphsSlide34
Slide35

Oriented graphsSlide36

Minimum inducibility (for tournaments)Slide37

3-graphsSlide38

Permutations (and permutons)

In our language, it is simply the theory of two linear

orderings on the same ground set and, as such, does

n

ot need any special treatment.

In fact, this is roughly the only other theory for which

s

emantics looks as nice as for

graphons. Slide39
Slide40

ConclusionMathematically 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?Slide41

Thank you