1. Extremal graph theory and limits of graphs László Lovász 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 2012 Kruskal-Katona Theorem (very special case): n k 3 Some old and new results from extremal graph theory

4. 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 4

5. September 2012 Theorem (Erdős): G contains no 4-cycles  #edges  n 3/2 /2 (Extremal: conjugacy graph of finite projective planes) 5 Some old and new results from extremal graph theory

6. September 2012 Theorem (Erdős-Stone-Simonovits):  (F)=3 6 Some old and new results from extremal graph theory

7. September 20127 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?

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 2012 Probability that random map V(F)  V(G) is a hom 9 Homomorphism functions Homomorphism: adjacency-preserving map If valid for large G, then valid for all

10. September 201210 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?

11. September 201211 Which inequalities between densities are valid? Undecidable… Hatami-Norine

12. September 2012 1 10 1/22/33/4 12 The main trick in the proof t(,G) – 2t(,G) + t(,G) = 0 …

13. September 201213 Which inequalities between densities are valid? Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy

14. September 201214 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?

15. 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 15 Which parameters are homomorphism functions?

16. September 2012 k=2:... M(f, k) 16 Connection matrices

17. September 2012 f = hom(.,H) for some weighted graph H  M(f,k) is positive semidefinite and has rank  c k Freedman - L - Schrijver Which parameters are homomorphism functions? 17

18. September 2012 k-labeled quantum graph: finite formal sum of k-labeled graphs 1 2 infinite dimensional linear space 18 Computing with graphs G k = {k-labeled quantum graphs}

19. September 2012 is a commutative algebra with unit element... Define products: 19 Computing with graphs G 1,G 2 : k-labeled graphs G 1 G 2 = G 1  G 2, labeled nodes identified

20. September 2012 Inner product: f: graph parameter extend linearly 20 Computing with graphs

21. September 2012 f is reflection positive Computing with graphs 21

22. September 2012 Write x ≥ 0 if hom(x,G) ≥ 0 for every graph G. Turán: -2+ Kruskal-Katona: - Blakley-Roy: - Computing with graphs 22

23. September 2012 -+- 2 =-+- - + - 2 +2 2 - =- + - -4 +2 Goodman’s Theorem Computing with graphs 23 + - 2  + ≥ 0 2 - = 2 -4-4 +2 t(,G) – 2t(,G) + t(,G) ≥ 0

24. September 2012 Question: Suppose that x ≥ 0. Does it follow that Positivstellensatz for graphs? 24 No! Hatami-Norine If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0.

25. September 2012 Let x be a quantum graph. Then x  0  A weak Positivstellensatz 25 L-Szegedy

26. the optimum of a semidefinite program is 0: minimize subject to M(f,k) positive semidefinite for all k f(K 1 )=1 f(G  K 1 )=f(G) September 2012 Proof of the weak Positivstellensatz (sketch 2 ) Apply Duality Theorem of semidefinite programming 26

27. September 201227 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?

28. 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 201228 Is there always an extremal graph? Quasirandom graphs

29. September 2012 Limit objects 29 (graphons)

30. 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 201230

31. September 2012 Limit objects 31 (graphons) t(F,W G )=t(F,G) (G 1,G 2,…) convergent:  F t(F,G n ) converges

32. For every convergent graph sequence (G n ) there is a graphon W such that G n  W. September 201232 Limit objects LS For every graphon W there is a graph sequence (G n ) such that G n  W. LS W is essentially unique (up to measure-preserving transformation). BCL

33. September 201233 Is there always an extremal graph? No, but there is always an extremal graphon. The space of graphons is compact.

34. September 2012 f = t(.,W)   k M(f,k) is positive semidefinite, f(  )=1 and f is multiplicative Semidefinite connection matrices 34 f: graph parameter

35. September 201235 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?

36. 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 201236 Extremal graphon problem

37. 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 201237 Finitely forcible graphons

38. Goodman 1/2 Graham- Chung- Wilson September 201238 Finitely forcible graphons

39. Stepfunctions  finite graphs with node and edgeweights Stepfunction: September 201239 Which graphs are extremal? Stepfunctions are finitely forcible L – V.T.Sós

40. d-regular graphon: d-regular September 201240 Finitely expressible properties

41. 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 201241 Finitely expressible properties

42. p(x,y)=0 p monotone decreasing symmetric polynomial finitely forcible ? September 201242 Finitely forcible graphons

43. S p(x,y)=0 Stokes September 201243 Finitely forcible graphons

44. Is the following graphon finitely forcible? angle <π/2 September 201244 Finitely forcible graphons

45. September 201245 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 ?

46. September 201246 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

47. September 201247 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 ?

48. September 201248 The local version Let Then t(F,W)  1.

49. September 201249 The idea of the proof 0 0<

50. September 201250 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).

51. September 201251 Common graphs Erdős: ? Thomason

52. September 201252 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs:  graph containing Jagger, Stovícek, Thomason

53. Common graphs September 201253

54. Common graphs September 201254 F common: is common. Franek-Rödl 8 +2 + +4 = 4 +2 +( +2 ) 2 +4( - )

55. Common graphs F locally common: September 201255 12 +3 +3 +12 + 12  2 +3  2 +3  4 +12  4 +  6 is locally common. Franek-Rödl

56. Common graphs September 201256  graph containing is locally common.  graph containing is locally common but not common. Not locally common:

57. Common graphs September 201257 F common:  - 1/2 1/2  - 1/2 1/2 8 +2 + +4 = 4 +2 +( -2 ) 2 is common. Franek-Rödl

58. September 201258 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs:  graph containing Jagger, Stovícek, Thomason