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

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"

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

May 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.

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

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

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

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

May 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

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

May 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)

May 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

May 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

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

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 are rational functions) Artin

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

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

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

May 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?

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

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

May 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?

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

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

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

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.

May 2012 A weak Positivstellensatz 27 L - Szegedy

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?

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

May 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

May 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?

May 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?

May 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

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 Finitely forcible graphons

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

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

p monotone decreasing symmetric polynomial finitely forcible ? January Finitely forcible graphons

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

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

May 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? ???