1. Which graphs are extremal? László Lovász Eötvös Loránd University, Budapest Joint work with Balázs Szegedy

2. Given: # of nodes n, # edges m, minimize # triangles. Types of extremal graphs Triangles

3. 1 10 Kruskal-Katona Bollobás 1/22/33/4 Razborov 2006 Mantel-Turán Goodman Fisher Lovász-Simonovits Types of extremal graphs Triangles

4. Types of extremal graphsTriangles Mantel (1907): extremal graph is Goodman (1959):extremal graph is Razborov (2006): extremal graph is Given: # of nodes n, # edges m, minimize # triangles.

5. Types of extremal graphsNonbipartite excluded subgraphs Erdős-Stone-Simonovits: Given: # of nodes n, excluded subgraphs F 1,...,F m, minimize # edges. Asymptotic extreme graph:, where

6. Given: # of nodes n, # edges m, minimize # quadrilaterals. Types of extremal graphsQuadrilaterals Asymptotically: Asymptotic extreme graph: random

7. Borgs, Chayes, L, Sós, B.Szegedy, Vesztergombi Limits of dense graph sequences Probability that random map V(G)  V(H) is a hom convergent: for every simple graph is convergent

8. graphons The limit object as a function

9. Adjacency matrix of graph G: Example 1: The limit object as a functionExamples Associated function W G :

10. The limit object as a functionExamples Stepfunctions  finite graphs with node and edgeweights Stepfunction:

11. Example 2: t(F,W)= 2 -|E(F)| # of eulerian orientations of F The limit object as a functionExamples

12. The limit object as a function

13. W is essentially unique (up to measure-preserving transformation). For every convergent graph sequence (G n ) there is a graphon such that Conversely, for every graphon W there is a graph sequence (G n ) such that Summary of main facts about graph limits

14. Extremal graphons Extremal graphon problem: Given find subject to

15. Finite forcing Graphon W is finitely forcible: Every finitely forcible graphon is extremal: minimize Every unique extremal graphon is finitely forcible. ??? Every extremal graph problem has a finitely forcible extremal graphon ???

16. Finite forcingExamples Goodman 1/2 Graham- Chung- Wilson

17. Finite forcingFinitely expressible properties d -regular graphon: d -regular

18. Finite forcing W is 0-1 valued, and can be rearranged to be monotone decreasing in both variables Finitely expressible properties “ W is 0-1 valued” is not finitely expressible. W is 0-1 valued

19. Finite forcingStepfunctions Every stepfunction is finitely forcible L – T.Sós Is the converse true?

20. Finite forcingOne-variable analogue Characterizations: inclusion-exclusion inequalities, semidefiniteness,... Characterizations: inclusion-exclusion inequalities, semidefiniteness,... The M ( k, f ) determine f up to measure preserving transformation The t ( F, W ) determine W up to measure preserving transformation ( F connected) t ( F 1, W ),... t ( F m, W ) are independent. M ( k 1, f ),... M ( k m, f ) are independent.

21. Finite forcingOne-variable analogue W stepfunction   finite number of subgraph densities that determine it. f stepfunction   finite number of moments that determine it. f not a stepfunction   k 1,..., k m  stepfunction g such that ?

22. Finite forcingSigns of polynomials p monotone decreasing symmetric polynomial finitely forcible ?

23. Finite forcingSigns of polynomials, proof sketch S p(x,y)=0

24. Is the following graphon finitely forcible? angle <π/2 Finite forcing2-dimensional graphons

25. Finite forcingUC graphons Union-complement graph (UC-graphs, cographs): Constructed from single nodes by disjoint union and complementation G is UC  no induced P 4 union and complement single node

26. Finite forcingUC graphons UC-graphon:... probability measure on paths connectednot connected UC-graphon is a stepfunction  tree is finite

27. Finite forcingRegular 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

28. Finite forcingRegular UC graphons  W is not a stepfunction

29. Finite forcingNon-forcible graphons W(x,y) is a polynomial  not finitely forcible ?? Every finitely forcible function has a finite range. ?? (stepfunction in a weaker sense)