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

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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

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September 20123 Some old and new results from extremal graph theory Probability that random map V(F) V(G) preserves edges Homomorphism: adjacency-preserving map

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September 20124 Some old and new results from extremal graph theory Theorem (Goodman): t(,G) – 2t(,G) + t(,G) ≥ 0 t(,G) = t(,G) 2

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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)

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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

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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

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September 20128 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?

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September 20129 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?

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September 201210 Which inequalities between densities are valid? If valid for large G, then valid for all

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April-May 201311 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 : rational functions) „Positivstellensatz” Artin

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September 201212 Which inequalities between densities are valid? Undecidable… Hatami-Norine

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September 2012 1 10 1/22/33/4 13 The main trick in the proof t(,G) – 2t(,G) + t(,G) = 0 …

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September 201214 Which inequalities between densities are valid? Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy

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September 201215 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?

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September 2012 Write a ≥ 0 if t(a,G) ≥ 0 for every graph G. Goodman: Computing with graphs 16 -2 + 0 Kruskal-Katona: - 0 Erdős: - 0

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September 2012 -+- 2 =-+- - + - 2 +2 2 - =- + - -4 +2 Computing with graphs 17 + - 2 2 - = + Goodman’s Theorem -2 + 0

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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.

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

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September 201220 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?

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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 201221 Is there always an extremal graph? Quasirandom graphs

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September 2012 Limit objects 22 (graphons)

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G 0 0 1 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 0 0 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 0 AGAG WGWG Graphs Graphons September 201223

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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

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A random graph with 100 nodes and with 2500 edges April-May 201325 Example: graph limit

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April-May 201326 A randomly grown uniform attachment graph on 200 nodes Example: graph limit

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April-May 201327 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

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

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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

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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

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September 201231 Is there always an extremal graph? No, but there is always an extremal graphon. The space of graphons is compact.

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September 201232 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?

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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 201233 Extremal graphon problem

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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 201234 Finitely forcible graphons

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Goodman 1/2 Graham- Chung- Wilson September 201235 Finitely forcible graphons

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Stepfunctions finite graphs with node and edgeweights Stepfunction: September 201236 Which graphs are extremal? Stepfunctions are finitely forcible L – V.T.Sós

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September 201237 Finitely forcible graphons

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Is the following graphon finitely forcible? angle <π/2 September 201238 Which graphons are finitely forcible?

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April-May 201339

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September 201240 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 ?

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September 201241 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

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September 201242 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 ?

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September 201243 The local version Let Then t(F,W) 1.

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September 201244 The idea of the proof 0 0<

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September 201245 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).

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September 201246 Common graphs Erdős: ? Thomason

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September 201247 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs: graph containing Jagger, Stovícek, Thomason

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Common graphs September 201248

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Common graphs September 201249 F common: is common. Franek-Rödl 8 +2 + +4 = 4 +2 +( +2 ) 2 +4( - )

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Common graphs F locally common: September 201250 12 +3 +3 +12 + 12 2 +3 2 +3 4 +12 4 + 6 is locally common. Franek-Rödl

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Common graphs September 201251 graph containing is locally common. graph containing is locally common but not common. Not locally common:

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Common graphs September 201252 F common: - 1/2 1/2 - 1/2 1/2 8 +2 + +4 = 4 +2 +( -2 ) 2 is common. Franek-Rödl

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September 201253 Common graphs F common: Hatami, Hladky, Kral, Norine, Razborov Common graphs: Sidorenko graphs (bipartite?) Non-common graphs: graph containing Jagger, Stovícek, Thomason

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

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September 2012 Graph parameter: isomorphism-invariant function on finite graphs k -labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes 1 2 55 Computing with graphs

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d-regular graphon: d-regular September 201256 Finitely expressible properties

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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 201257 Finitely expressible properties

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