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

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Presentation on theme: "Which graphs are extremal? László Lovász Eötvös Loránd University Budapest September 20121."— Presentation transcript:

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 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 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 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 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 Which inequalities between densities are valid? If valid for large G, then valid for all

11 April-May 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 r m 2 (r 1,...,r m : rational functions) „Positivstellensatz” Artin

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

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

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

15 September 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  0 Kruskal-Katona: -  0 Erdős: -  0

17 September = = Computing with graphs  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 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 Is there always an extremal graph? Quasirandom graphs

22 September 2012 Limit objects 22 (graphons)

23 G AGAG WGWG Graphs  Graphons September

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 Example: graph limit

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

27 April-May 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 Is there always an extremal graph? No, but there is always an extremal graphon. The space of graphons is compact.

32 September 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 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 Finitely forcible graphons

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

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

37 September Finitely forcible graphons

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

39 April-May

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

44 September The idea of the proof 0 0<

45 September 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 Common graphs Erdős: ? Thomason

47 September 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

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

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

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

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

53 September 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 Computing with graphs

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


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