1. Dense graph limit theory: Extremal graph theory László Lovász Eötvös Loránd University, Budapest May 20121

2. 2 Recall some math t(F,G): Probability that random map V(F)  V(G) preserves edges (G 1,G 2,…) convergent:  F t(F,G n ) is convergent W 0 = { W: [0,1] 2  [0,1], symmetric, measurable } G n  W :  F: t(F,G n )  t(F,W) "graphon"

3. May 20123 Recall some math The semimetric space ( W 0,   ) is compact.

4. May 20124 Recall some math For every convergent graph sequence (G n ) there is a W  W 0 such that G n  W. W is essentially unique (up to measure-preserving transformation). Conversely,  W  (G n ) such that G n  W.

5. May 2012 Turán’s Theorem (special case proved by Mantel): G contains no triangles  #edges  n 2 /4 Theorem (Goodman): Extremal: 5 A sampler of results from extremal graph theory

6. May 2012 Kruskal-Katona Theorem (very special case): n k 6 Some old and new results from extremal graph theory

7. May 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 7

8. May 2012 Theorem (Erdős): G contains no 4-cycles  #edges  n 3/2 /2 (Extremal: conjugacy graph of finite projective planes) 8 Some old and new results from extremal graph theory Cauchy-Schwarz twice

9. May 20129 Thomason, Chung-Graham-Wilson Common properties of random graphs  (n,p) (n  ): (1)almost all degrees  pn, almost all codegrees  p 2 n. (2)for X,Y  V(G), e(X,Y)= p|X||Y|+o(n 2 ) (3)for every graph F, t(F,  )  p |E(F)| (4) t( |,  )  p, t( ,  )  p 4 For any sequence of graphs G n (|V(G n )|=n  ), these properties are equivalent. Quasirandom graph sequences

10. Example: Paley graphs p : prime  1 mod 4 Quasirandom graph sequences xy  E(G)  x-y = 

11. May 201211 Quasirandom graph sequences For every graph F, t(F,G n )  p |E(F)|  G n  p For every graph F, t(F,  )  p |E(F)|  t( |,  )  p, t( ,  )  p 4 If t( |, W ) = p, t( , W ) =p 4, then W  p (equality in Cauchy-Schwarz)

12. May 201212 General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Local vs. global extrema

13. May 201213 General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Local vs. global extrema

14. May 201214 Extremal problems If valid asymptotically for large G, then valid for all

15. May 201215 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 are rational functions) Artin

16. May 201216 Which inequalities between densities are valid? Undecidable… Hatami-Norine

17. May 2012 1 10 1/22/33/4 17 The main trick in the proof t(,G) – 2t(,G) + t(,G) = 0 …

18. May 201218 Which inequalities between densities are valid? Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy

19. May 201219 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?

20. May 2012 Write x ≥ 0 if hom(x,G) =  x G G ≥ 0 for every graph G. Turán: -2+ Kruskal-Katona: - Computing with graphs 20 Erdős: -  formal linear combination of graphs

21. May 2012 -+- 2 =-+- - + - 2 +2 2 - =- + - -4 +2 Goodman’s Theorem Computing with graphs 21 + - 2  + ≥ 0 2 - = 2 -4-4 +2 t(,G) – 2t(,G) + t(,G) ≥ 0

22. May 2012 Graph parameter: isomorphism-invariant function on finite graphs k -labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes 1 2 22 Which parameters are homomorphism functions?

23. May 2012 k=2:... M(f, k) 23 Connection matrices

24. May 2012 Freedman - L - Schrijver is positive semidefinite and has rank  c k. Which parameters are homomorphism functions? 24

25. May 2012 is positive semidefinite, f ( )=1 and f is multiplicative 25 L - Szegedy Which parameters are homomorphism functions?

26. May 2012 Question: Suppose that x ≥ 0. Does it follow that Positivstellensatz for graphs? 26 No! Hatami-Norine If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0.

27. May 2012 A weak Positivstellensatz 27 L - Szegedy

28. November 2010 Semidefinite formulation of the Mantel-Turán Theorem 28 G: (large) unknown graph x F = t(F,G): variables Can be ignored Infinitely many variables must be cut to finite size arbitrarily small error?

29. The optimum of the semidefinite program minimize subject to M(x,k) positive semidefinite for all k =1 is 0. May 2012 Proof of the weak Positivstellensatz (sketch 2 ) Apply Duality Theorem of semidefinite programming. 29

30. May 201230 General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Local vs. global extrema

31. May 201231 Alon-Stav If P is a hereditary graph property, then there is a 0≤p≤1 such that G(n,p) is asymptotically farthest from P among all n-node graphs. in “edit distance” Local optimum: when is it global?

32. May 201232 Want: maximize d 1 (U, R ): U  K K is convex K is invariant under measure preserving transformations d 1 (., R ) is concave on K d 1 is maximized on K by a constant function Local optimum: when is it global?

33. May 201233 General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Which graphs are extremal? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Local vs. global extrema

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 ??? May 201234 Finitely forcible graphons

35. Goodman 1/2 Graham- Chung- Wilson May 201235 Many finitely forcible graphons

36. Stepfunctions  finite graphs with node and edgeweights Stepfunction: May 201236 L – V.T.Sós Many finitely forcible graphons

37. p monotone decreasing symmetric polynomial finitely forcible ? January 201137 Finitely forcible graphons

38. S p(x,y)=0 Stokes January 201138 Finitely forcible graphons

39. May 201239 Not too many finitely forcible graphons Finitely forcible graphons form a set of first category in ( W 0,   ).

40. May 201240 Finitely forcible graphons: conjectures ??? Finitely forcible  space of “rows” has finite dimension ??? ??? Finitely forcible  algebra of k-labeled quantum graphs mod W is finitely generated ??? W=1 iff angle <π/2 ??? Is this graphon finitely forcible? ???