1. Which graphs are extremal? László Lovász Eötvös Loránd University Budapest September 20121

2. Turán’s Theorem (special case proved by Mantel): G contains no triangles  #edges  n 2 /4 Theorem (Goodman): Extremal: 2 Some old and new results from extremal graph theory

3. September 20123 Some old and new results from extremal graph theory Probability that random map V(F)  V(G) preserves edges Homomorphism: adjacency-preserving map

4. September 20124 Some old and new results from extremal graph theory Theorem (Goodman): t(,G) – 2t(,G) + t(,G) ≥ 0 t(,G) = t(,G) 2

5. September 2012 Kruskal-Katona Theorem (very special case): n k 5 Some old and new results from extremal graph theory t(,G) 2 ≥ t(,G) 3 t(,G) ≥ t(,G)

6. September 2012 Semidefiniteness and extremal graph theoryTricky examples 1 10 Kruskal-Katona Bollobás 1/22/33/4 Razborov 2006 Mantel-Turán Goodman Fisher Lovász-Simonovits Some old and new results from extremal graph theory 6

7. September 2012 Theorem (Erdős): G contains no 4-cycles  #edges  n 3/2 /2 (Extremal: conjugacy graph of finite projective planes) 7 Some old and new results from extremal graph theory t(,G) ≥ t(,G) 4

8. September 20128 General questions about extremal graphs - Is there always an extremal graph? -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz?

9. September 20129 General questions about extremal graphs - Is there always an extremal graph? -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz?

10. September 201210 Which inequalities between densities are valid? If valid for large G, then valid for all

11. April-May 201311 Analogy with polynomials p(x 1,...,x n )  0 for all x 1,...,x n   undecidable Matiyasevich for all x 1,...,x n   decidable Tarski  p = r 1 2 +...+ r m 2 (r 1,...,r m : rational functions) „Positivstellensatz” Artin

12. September 201212 Which inequalities between densities are valid? Undecidable… Hatami-Norine

13. September 2012 1 10 1/22/33/4 13 The main trick in the proof t(,G) – 2t(,G) + t(,G) = 0 …

14. September 201214 Which inequalities between densities are valid? Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy

15. September 201215 General questions about extremal graphs - Is there always an extremal graph? -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz?

16. September 2012 Write a ≥ 0 if t(a,G) ≥ 0 for every graph G. Goodman: Computing with graphs 16 -2 +  0 Kruskal-Katona: -  0 Erdős: -  0

17. September 2012 -+- 2 =-+- - + - 2 +2 2 - =- + - -4 +2 Computing with graphs 17 + - 2  2 - = + Goodman’s Theorem -2 +  0

18. September 2012 Question: Suppose that x ≥ 0. Does it follow that Positivstellensatz for graphs? 18 No! Hatami-Norine If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0.

19. September 2012 A weak Positivstellensatz 19 L - Szegedy (ignoring labels and isolated nodes)

20. September 201220 General questions about extremal graphs - Is there always an extremal graph? -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz?

21. Minimize over x  0 minimum is not attained in rationals Minimize t(C 4,G) over graphs with edge-density 1/2 minimum is not attained among graphs always >1/16, arbitrarily close for random graphs Real numbers are useful Graph limits are useful September 201221 Is there always an extremal graph? Quasirandom graphs

22. September 2012 Limit objects 22 (graphons)

23. G 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 AGAG WGWG Graphs  Graphons September 201223

24. September 2012 Limit objects 24 (graphons) t(F,W G )=t(F,G) (G 1,G 2,…) convergent:  F t(F,G n ) converges Borgs-Chayes-L-Sós-Vesztergombi

25. A random graph with 100 nodes and with 2500 edges April-May 201325 Example: graph limit

26. April-May 201326 A randomly grown uniform attachment graph on 200 nodes Example: graph limit

27. April-May 201327 Limit objects: the math For every convergent graph sequence (G n ) there is a W  W 0 such that G n  W. Conversely,  W  (G n ) such that G n  W. L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs-Chayes-L

28. September 2012 k=2:... M(f, k) 28 Connection matrices

29. September 2012  W: f = t(.,W)   k M(f,k) is positive semidefinite, f(  )=1 and f is multiplicative Semidefinite connection matrices 29 f: graph parameter L-Szegedy

30. the optimum of a semidefinite program is 0: minimize subject to M(x,k) positive semidefinite  k x(K 0 )=1 x(G  K 1 )=x(G) September 2012 Proof of the weak Positivstellensatz (sketch 2 ) Apply Duality Theorem of semidefinite programming 30

31. September 201231 Is there always an extremal graph? No, but there is always an extremal graphon. The space of graphons is compact.

32. September 201232 General questions about extremal graphs - Is there always an extremal graph? -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz?

33. Given quantum graphs g 0,g 1,…,g m, find max t(g 0,W) subject to t(g 1,W) = 0 … t(g m,W) = 0 September 201233 Extremal graphon problem

34. 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 ?? September 201234 Finitely forcible graphons

35. Goodman 1/2 Graham- Chung- Wilson September 201235 Finitely forcible graphons

36. Stepfunctions  finite graphs with node and edgeweights Stepfunction: September 201236 Which graphs are extremal? Stepfunctions are finitely forcible L – V.T.Sós

37. September 201237 Finitely forcible graphons

38. Is the following graphon finitely forcible? angle <π/2 September 201238 Which graphons are finitely forcible?

39. April-May 201339

40. September 201240 The Simonovits-Sidorenko Conjecture F bipartite, G arbitrary  t(F,G) ≥ t(K 2,G) |E(F)| Known when F is a tree, cycle, complete bipartite… Sidorenko F is hypercube Hatami F has a node connected to all nodes in the other color class Conlon,Fox,Sudakov F is "composable" Li, Szegedy ?

41. September 201241 The Simonovits-Sidorenko Conjecture Two extremal problems in one: For fixed G and |E(F)|, t(F,G) is minimized by F= … asymptotically For fixed F and t(,G), t(F,G) is minimized by random G

42. September 201242 The integral version Let W  W 0, W≥0, ∫ W=1. Let F be bipartite. Then t(F,W)≥1. For fixed F, t(F,W) is minimized over W≥0, ∫ W=1 by W  1 ?

43. September 201243 The local version Let Then t(F,W)  1.

44. September 201244 The idea of the proof 0 0<

45. September 201245 The idea of the proof Main Lemma: If -1≤ U ≤ 1, shortest cycle in F is C 2r, then t(F,U) ≤ t(C 2r,U).

46. September 201246 Common graphs Erdős: ? Thomason

47. September 201247 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs:  graph containing Jagger, Stovícek, Thomason

48. Common graphs September 201248

49. Common graphs September 201249 F common: is common. Franek-Rödl 8 +2 + +4 = 4 +2 +( +2 ) 2 +4( - )

50. Common graphs F locally common: September 201250 12 +3 +3 +12 + 12  2 +3  2 +3  4 +12  4 +  6 is locally common. Franek-Rödl

51. Common graphs September 201251  graph containing is locally common.  graph containing is locally common but not common. Not locally common:

52. Common graphs September 201252 F common:  - 1/2 1/2  - 1/2 1/2 8 +2 + +4 = 4 +2 +( -2 ) 2 is common. Franek-Rödl

53. September 201253 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs:  graph containing Jagger, Stovícek, Thomason

54. September 2012 Theorem (Erdős-Stone-Simonovits):  (F)=3 54 Some old and new results from extremal graph theory

55. September 2012 Graph parameter: isomorphism-invariant function on finite graphs k -labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes 1 2 55 Computing with graphs

56. d-regular graphon: d-regular September 201256 Finitely expressible properties

57. W is 0-1 valued, and can be rearranged to be monotone decreasing in both variables "W is 0-1 valued" is not finitely expressible in terms of simple gaphs. W is 0-1 valued September 201257 Finitely expressible properties