Which graphs are extremal?

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Which graphs are extremal?
László Lovász Eötvös Loránd University, Budapest
Joint work with Balázs Szegedy
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Types of extremal graphs
Triangles
Given of nodes n, edges m, minimize
triangles.
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Types of extremal graphs
Triangles
1
1
0
1/2
2/3
3/4
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Types of extremal graphs
Triangles
Given of nodes n, edges m, minimize
triangles.
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Types of extremal graphs
Nonbipartite excluded subgraphs
Given of nodes n, excluded subgraphs
F1,...,Fm, minimize edges .
Erdos-Stone-Simonovits
Asymptotic extreme graph ,
where
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Types of extremal graphs
Quadrilaterals
Given of nodes n, edges m, minimize
quadrilaterals.
Asymptotic extreme graph random
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Limits of dense graph sequences
Borgs, Chayes, L, Sós, B.Szegedy, Vesztergombi
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The limit object as a function
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The limit object as a function
Examples
Example 1
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The limit object as a function
Examples
Stepfunction
Stepfunctions ? finite graphs with node and
edgeweights
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The limit object as a function
Examples
Example 2
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The limit object as a function
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Summary of main facts about graph limits
W is essentially unique (up to measure-preservin
g transformation).
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Extremal graphons
Extremal graphon problem Given find subj
ect to
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Finite forcing
Graphon W is finitely forcible
Every unique extremal graphon is finitely
forcible.
??? Every extremal graph problem has a finitely
forcible extremal graphon ???
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Finite forcing
Examples
Goodman
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Finite forcing
Finitely expressible properties
d-regular graphon
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Finite forcing
Finitely expressible properties
W is 0-1 valued is not finitely expressible.
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Finite forcing
Stepfunctions
L T.Sós
Is the converse true?
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Finite forcing
One-variable analogue
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Finite forcing
One-variable analogue
?
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Finite forcing
Signs of polynomials
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Finite forcing
Signs of polynomials, proof sketch
S
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Finite forcing
2-dimensional graphons
Is the following graphon finitely forcible?
angle ltp/2
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Finite forcing
UC graphons
Union-complement graph (UC-graphs, cographs)
Constructed from single nodes
by disjoint union and
complementation
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Finite forcing
UC graphons
...
...
...
...
UC-graphon is a stepfunction ? tree is finite
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Finite forcing
Regular UC graphons
UC-graphon is regular ? tree is locally finite
For every locally finite tree with outdegrees 2
there is a unique regular UC-graphon.
d-regular UC-graphon is a stepfunction ? d is
rational
?0 d 1 ?d-regular UC-graphon
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Finite forcing
Regular UC graphons
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Finite forcing
Non-forcible graphons
?? Every finitely forcible function has a finite
range. ?? (stepfunction in a weaker
sense)