Extremal graph theory and limits of graphs László Lovász September 20121

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

September 2012 Kruskal-Katona Theorem (very special case): n k 3 Some old and new results from extremal graph theory

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

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

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

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?

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?

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

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?

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

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

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

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?

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

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

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

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}

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

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

September 2012 f is reflection positive Computing with graphs 21

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

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

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.

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

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

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?

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

September 2012 Limit objects 29 (graphons)

G AGAG WGWG Graphs  Graphons September

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

For every convergent graph sequence (G n ) there is a graphon W such that G n  W. September 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

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

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

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?

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

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

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

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

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

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

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

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

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

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 ?

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

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 ?

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

September The idea of the proof 0 0<

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).

September Common graphs Erdős: ? Thomason

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

Common graphs September

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

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

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

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

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