Which graphs are extremal? László Lovász Eötvös Loránd University Budapest 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 Some old and new results from extremal graph theory Probability that random map V(F)  V(G) preserves edges Homomorphism: adjacency-preserving map

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

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)

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

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

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

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

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

September /22/33/4 13 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 Write a ≥ 0 if t(a,G) ≥ 0 for every graph G. Goodman: Computing with graphs  0 Kruskal-Katona: -  0 Erdős: -  0

September = = Computing with graphs  2 - = + Goodman’s Theorem -2 +  0

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.

September 2012 A weak Positivstellensatz 19 L - Szegedy (ignoring labels and isolated nodes)

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 22 (graphons)

G AGAG WGWG Graphs  Graphons September

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

A random graph with 100 nodes and with 2500 edges April-May Example: graph limit

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

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

September 2012 k=2:... M(f, k) 28 Connection matrices

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

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

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

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

September Finitely forcible graphons

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

April-May

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

September 2012 Theorem (Erdős-Stone-Simonovits):  (F)=3 54 Some old and new results from extremal graph theory

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

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