Which graphs are extremal?

1. Which graphs are extremal? LászlóLovász Eötvös Loránd University Budapest

2. Some old and new results from extremal graph theory Extremal: Theorem (Goodman): Turán’s Theorem (special case proved by Mantel): G contains no triangles  #edgesn2/4

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

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

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

6. Some old and new results from extremal graph theory Kruskal-Katona Razborov 2006 Fisher Goodman Bollobás Mantel-Turán Lovász-Simonovits Semidefiniteness and extremal graph theory Tricky examples 1 0 1/2 2/3 3/4 1

7. Some old and new results from extremal graph theory Theorem (Erdős): G contains no 4-cycles  #edgesn3/2/2 (Extremal: conjugacy graph of finite projective planes) t( ,G) ≥ t( ,G)4

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

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

10. Which inequalities between densities are valid? IfvalidforlargeG, thenvalidforall

11. Analogy with polynomials p(x1,...,xn)0 for all x1,...,xnRdecidable Tarski • p = r12 + ...+rm2 (r1, ...,rm:rational functions) • „Positivstellensatz”Artin for all x1,...,xnZundecidable Matiyasevich

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

13. The main trick in the proof 1 t( ,G) – 2t( ,G) + t( ,G) = 0 0 1/2 2/3 3/4 1 …

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

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

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

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

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

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

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

21. Is there always an extremal graph? Minimize over x0 always >1/16, arbitrarily close for random graphs Real numbers are useful minimum is not attained in rationals Minimize t(C4,G) over graphs with edge-density 1/2 Quasirandom graphs Graph limits are useful minimum is not attained among graphs

22. Limit objects (graphons)

23. Graphs  Graphons 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 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 G AG WG

24. Limit objects (graphons) t(F,WG)=t(F,G) (G1,G2,…) convergent: Ft(F,Gn) converges Borgs-Chayes-L-Sós-Vesztergombi

25. Example: graph limit A random graphwith 100 nodes and with 2500 edges

26. Example: graph limit A randomly grown uniform attachment graphon200 nodes

27. Limit objects: themath For every convergent graph sequence (Gn) there is a WW0 such thatGnW. Conversely, W(Gn) such thatGnW. L-Szegedy W is essentially unique (up to measure-preserving transformation). Borgs-Chayes-L

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

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

30. Proof of the weak Positivstellensatz (sketch2) the optimum of a semidefinite program is 0: minimize subject to M(x,k)positive semidefinite k x(K0)=1 x(GK1)=x(G) Apply Duality Theorem of semidefinite programming

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

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

33. Extremal graphon problem Given quantum graphs g0,g1,…,gm, find max t(g0,W) subject to t(g1,W) = 0 … t(gm,W) = 0

34. Finitely forcible graphons Every finitely forcible graphon is extremal: minimize Finite forcing GraphonW is finitely forcible: Every unique extremal graphon is finitely forcible. ??Every extremal graph problem has a finitely forcible extremal graphon ??

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

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

37. Finitely forcible graphons

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

39. Thanks, that’s all for today!

40. The Simonovits-Sidorenko Conjecture ? F bipartite, G arbitrary t(F,G) ≥ t(K2,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. The Simonovits-Sidorenko Conjecture Two extremal problems in one: For fixed G and |E(F)|, t(F,G) is minimized by F= For fixed F and t( ,G), t(F,G) is minimized by random G asymptotically …

42. The integral version Let WW0, 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. The local version Let Then t(F,W)  1.

44. The idea of the proof 0< 0

45. The idea of the proof Main Lemma: If -1≤ U ≤ 1, shortest cycle in F is C2r, then t(F,U) ≤ t(C2r,U).

46. Common graphs ? Erdős: Thomason

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

48. Common graphs

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

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