Elusive problems in extremal graph theory

Transcription

1 Elusive problems in extremal graph theory Andrzej Grzesik (Krakow) Dan Král (Warwick) László Miklós Lovász (MIT/Stanford) 2/5/27

2 Overview of talk uniqueness of extremal configurations motivation and formulation of problem graph limits representation of large graphs finitely forcible graph limits large graphs with assymptotically unique structures main result, proof tools and extensions 2

3 Turán Problems Maximum edge-density of H-free graph Mantel s Theorem (97): 2 for H = K 3 (Kn 2,n 2 ) Turán s Theorem (94): l 2 l for H = K l (K n l,..., n l ) Erdős-Stone Theorem (946): χ(h) 2 χ(h) extremal examples unique up to o(n 2 ) edges 3

4 Edge vs. Triangle Problem Minimum density of K 3 for a specific edge-density determined by Razborov (28), K αn,...,αn,( kα)n extensions by Nikiforov (2) and Reiher (26) for K l Pikhurko and Razborov (27) gave extremal examples generally not unique, can be made unique by K n = 4

5 Another example Minimum sum of densities of K 3 and K 3 Goodman s Bound (959): K 3 +K 3 4 every n/2-regular graph is a minimizer minimizer can be made unique K 3 =, or K 3 =, or C 4 = /6 (Erdős-Rényi random graph G n,/2 ) 5

6 This talk Conjecture (Lovász 28, Lovász and Szegedy 2) Every finite feasible set H i = d i, i =,...,k, can be extended to a finite feasible set with an asymptotically unique structure. Every extremal problem has a finitely forcible optimum. Theorem (Grzesik, K., Lovász Jr.): FALSE 6

7 Graph limits large networks large graphs how to represent? how to model? how to generate? concise (analytic) representation of large graphs we implicitly use limits in our considerations anyway mathematics motivation extremal graph theory What is a typical structure of an extremal graph? calculations avoiding smaller order terms in this talk: dense graphs ( E = Ω( V 2 )) Borgs, Chayes, Lovász, Sós, Szegedy, Vesztergombi,... convergence vs. analytic representation 7

8 Convergent graph sequence d(h,g) = probability H -vertex subgraph of G is H a sequence (G n ) n N of graphs is convergent if d(h,g n ) converges for every H examples: K n, K αn,n, blow ups G[K n ] Erdős-Rényi random graphs G n,p, planar graphs extendable to other discrete structures 8

9 Limit object: graphon graphon W : [,] 2 [,], s.t. W(x,y) = W(y,x) W-random graph of order n random points x i [,], edge probability W(x i,x j ) d(h,w) = prob. H -vertex W-random graph is H W is a limit of (G n ) n N if d(h,w) = lim n d(h,g n) 9

10 Limit object: graphon graphon W : [,] 2 [,], s.t. W(x,y) = W(y,x) W-random graph of order n random points x i [,], edge probability W(x i,x j ) d(h,w) = prob. H -vertex W-random graph is H W is a limit of (G n ) n N if d(h,w) = lim n d(h,g n) every convergent sequence of graphs has a limit W-random graphs converge to W with probability one

11 Applications of graph limits extremal combinatorics flag algebras of Razborov density calculations, computer search computer science property and parameter testing cover of the space of all graphons structure of typical graphs graphon entropy, number of graphs c (n 2)

12 Statement of problem Conjecture (Lovász 28, Lovász and Szegedy 2): Every finite feasible set H i = d i, i =,...,k, can be extended to a finite feasible set that is satisifed by a unique graphon. uniqueness of graphons (Borgs, Chayes, Lovász 2) W(x,y) and W ϕ (x,y) := W(ϕ(x),ϕ(y)) are the same A graphon W is finitely forcible if there exist H,...,H k and d,...,d k such that W is the only graphon with the density of H i equal to d i. 2

13 Finitely forcible graph limits Lovász, Sós (28): Step graphons are finitely forcible. extremal graph theory problem finitely forcible optimal solution simple structure gives new bounds on old problems Conjectures (Lovász and Szegedy): The space T(W) of a finitely forcible W is compact. The space T(W) has finite dimension. 3

14 Finitely forcible graphons Theorem (Glebov, K., Volec): T(W) can fail to be locally compact Theorem (Glebov, Klimošová, K.): T(W) can have a part homeomorphic to [,] Theorem (Cooper, Kaiser, K., Noel): finitely forcible W such that every ε-regular partition has at least 2 ε 2 /loglogε parts (for inf. many ε ). Theorem (Cooper, K., Martins): Every graphon is a subgraphon of a finitely forcible graphon. 4

15 Rademacher graphon 5

16 Non-regular graphon A A B C D E F G P Q R B C D E F G P Q R 6

17 Universal construction A B C D E F G P Q R A B C D E F G W F P Q R 7

18 Main result Theorem (Grzesik, K., Lovász Jr.) graphon family W, graphs H i, reals d i, i =,...,m W W d(h i,w) = d i for i =,...,m no graphon in W is finitely forcible A B C A B C D A D B D C D D D E D F D G E F G H D A D B D C D D D E D F D G E F G H 8

19 Some details of the proof graphons W P ( z), z [,] N z satisfies polynomial inequalities in P (e.g. z +z 2 2 ) z constrained to be from Z [,] N such that d(h,w P ( z)) = f (z,z 2 ) d(h 2,W P ( z)) = f 2 (z,z 2, z 3,z 4,z 5 ) d(h 3,W P ( z)) = f 3 (z,z 2, z 3,z 4,z 5, z 6,z 7,z 8,z 9 ) the set Z is non-trivial there exists a bijective map from [,] N to Z such that (x ) (z,z 2 ), (x,x 2 ) (z,z 2, z 3,z 4,z 5 ), etc. 9

20 Some details of the proof graphons W P ( z), z [,] N z satisfies polynomial inequalities in P (e.g. z +z 2 2 ) independent of P: there exist graphs H,...,H k there exist polynomials q,...,q l in d(h i,w) for every P: there exist reals α,...,α l W P ( z) are precisely graphons satisfying q i = α i analysis of the dependance of d(h i,w P ( z)) on P approximation of inverse maps by polyn. inequalities 2

21 Possible extensions techniques universal to prove more general results equalize other functions than subgraph densities Theorem (Grzesik, K., Lovász Jr.) graphon family W, graphs H i, reals d i, i =,...,m W W d(h i,w) = d i for i =,...,m no graphon in W is finitely forcible all graphons in W have the same entropy extremal problems with no typical structure 2

22 Thank you for your attention! 22