Flag Algebra Methods (more formal approach) Bernard Lidick y 6th - PowerPoint PPT Presentation

Flag Algebra Methods (more formal approach) Bernard Lidick´ y 6th Lake Michigan Workshop on Combinatorics and Graph Theory Apr 7, 2019

Outline • Flag Algebras “definitions” • First try for Mantel’s theorem • More automatic approach • Additional constraints • maybe break • Define flag algebras • Graph sequences and homomorphisms • Tur´ an’s Theorem (in limit) • Finally Mantel’s Theorem (for real) • mega break • Applications 2

Building an algebra Let F denote the set of all graphs (up to isomorphism). Let R F be the set of all finite formal linear combinations of graphs. � � − 1 + 2 2 · 4 · 3 · 3 · + π It comes with natural addition and multiplication by a real number (think of a vector ∈ R F .) How to define multiplication of two elements in R F ? 3

Towards multiplication Let F denote the set of all graphs. Let F 1 , F 2 , F ∈ F such that | F 1 | + | F 2 | ≤ | F | . Definition Let X 1 , X 2 ⊆ V ( F ) be random disjoint of sizes | F 1 | and | F 2 | . P ( F 1 , F 2 ; F ) is the probability F [ X i ] ∼ = F i for both i . � � = 2 � � = 1 P , ; P , ; 3 2 � � � � = 1 = 1 P , ; P , ; 6 6 4

Towards multiplication Let F denote the set of all graphs. Let F 1 , F 2 , F ∈ F such that | F 1 | + | F 2 | ≤ | F | . Definition Let X 1 , X 2 ⊆ V ( F ) be random disjoint of sizes | F 1 | and | F 2 | . P ( F 1 , F 2 ; F ) is the probability F [ X i ] ∼ = F i for both i . � � = 2 � � = 1 P , ; P , ; 3 2 � � � � = 1 = 1 − 2 P , ; P , ; 6 6 4

Multiplication Let F denote the set of all graphs. Let F 1 , F 2 ∈ F . We need F 1 · F 2 ∈ R F . Define � F 1 · F 2 := P ( F 1 , F 2 ; F ) F , F ∈F ℓ where | F 1 | + | F 2 | = ℓ . Notice there is NO + o (1). � � � � · = P , ; · + P , ; · + 1 + 1 + 1 + 1 = 0 + · · · 6 3 2 2 This extends to a · b for a , b ∈ R F . 5

Factorizing Recall from previous presentation we had identities like + 1 + 2 = 0 + 1 3 3 Let K be linear subspace generated by � P ( F , F ′ ) F ′ F − (= 0) , F ′ ∈F ℓ where F ∈ F and | F | ≤ ℓ . Algebra A is R F factorized by K . a , b ∈ R F are in one equivalence class if a = b + c for some c ∈ K Note: Think of R F and A as Z × Z and Q . (correctness of definitions proved by Razborov) 6

Algebra A • F is the set of all graphs. • F ℓ is on ℓ vertices. • R F formal linear combinations • K := span ( F − � F ′ ∈F ℓ P ( F , F ′ ) F ′ ) • A is R F factorized by K . • addition in A comes from R F • F 1 · F 2 = � F ∈ F ℓ P ( F 1 , F 2 ; F ) F , • F ∈ F is called a flag . • informally called unlabeled flags 7

Convergent Graph Sequence Let F denote the set of all graphs. Definition A sequence of graphs ( G n ) n ∈ N is convergent if for every finite graph H , lim n →∞ P ( H , G n ) exists. Examples: � if H ∼ 1 = K m n →∞ P ( H , K n ) = lim 0 otherwise  if H ∼ 1 / 2 = K 2    if H ∼ 3 / 4 = P 2 n →∞ P ( H , K n , n ) = lim . .   .  � 1 is | E ( H ) | = 0 n →∞ P ( H , P n ) = lim 0 otherwise Gives map F → [0 , 1] or a point in [0 , 1] F . 8

Convergent Graph Sequence Let F denote the set of all graphs. Definition A sequence of graphs ( G n ) n ∈ N is convergent if for every finite graph H , lim n →∞ P ( H , G n ) exists. For all H ∈ F n →∞ P ( H , K n , n ) = lim lim n →∞ P ( H , K n , n ∪ K 1 ) = lim n →∞ P ( H , K n , n + K 1 ) � � In particular for H ∈ , . Small changes in ( G n ) n ∈ N are not noticeable. 9

Positive Homomorphisms Let Hom ( A , R ) be the set of all homomorphisms from A to R . I.e. for any φ ∈ Hom ( A , R ) and a , b ∈ A : • φ ( a + b ) = φ ( a ) + φ ( b ) • φ ( a · b ) = φ ( a ) · φ ( b ). Since p ( H , G ) ∈ [0 , 1], we consider only Hom + ( A , R ), i.e. φ ( H ) ≥ 0 for all H ∈ F Notice φ ( ∅ ) = 1. Theorem (Razborov) Hom + ( A , R ) corresponds exactly to the convergent sequences. Theorem (Felix; Podolski; (only for graphs)) Let a ∈ R F . φ ( a ) = 0 for all φ ∈ Hom + ( A , R ) iff a ∈ K . 10

Theorem (Mantel) Every triangle-free graph on n vertices has at most n 2 / 4 edges. Theorem (Mantel from previous presentation) ≤ 1 For a large graph if = 0 then 2 + o (1) . Theorem (Mantel with Homomorphisms) For every φ ∈ Hom + ( A , R ) holds that � � ≤ 1 � � if φ = 0 then φ 2 . 11

Example from last time 0 ≤ 3 · − − + 3 · + o (1) We want to find a ∈ A such that for EVERY φ ∈ Hom + ( A , R ) we have φ ( a ) ≥ 0. We write a ≥ 0. Theorem (Hatami and Norin 11) Determining if a ≥ 0 is not algorithmically decidable. Norin: “Extremal combinatorics remains an art”. But we can still generate a lot of them! 12

Algebra A σ • vertices of σ ∈ F are labeled by 1 , . . . , | σ | . • F σ is the set of all graphs each containing a fixed induced labeled copy of σ . • F σ ℓ is on ℓ vertices. 2 2 • R F σ formal linear combinations 1 1 • K σ := span ( F − � ℓ P ( F , F ′ ) F ′ ) F ′ ∈F σ • A σ is R F σ factorized by K σ . • addition in A σ comes from R F σ 2 • F 1 · F 2 = � ℓ P ( F 1 , F 2 ; F ) F , 1 F ∈ F σ • F ∈ F σ is called a σ -flag . 3 • σ is called a type 2 1 13

F 1 · F 2 = � ℓ P ( F 1 , F 2 ; F ) F F ∈ F σ · · · X 2 X 1 1 2 | σ | F Pick randomly X 1 , X 2 ⊂ V ( F ) such that X 1 ∩ X 2 are exactly all labeled vertices and | X i | = | F i | . P ( F 1 , F 2 ; F ) is the probability that X 1 ∼ = F 1 and X 2 ∼ = F 2 . 14

Averaging (unlabeling) operator Let F = ( G , θ ) be a σ -flag, where θ : 1 , . . . , | σ | → V ( G ). Define � F � σ = q θ ( F ) · G , where q θ ( F ) is the probability that ( G , θ ′ ) is isomorphic to F for a random injective θ ′ : 1 , . . . , | σ | → V ( G ). � � 2 = 6 � � 20 1 σ Linear extension gives the averaging operator � · � σ : R F σ → R F . It is a linear mapping, not a homomorphism. � F 1 + F 2 � σ = � F 1 � σ + � F 2 � σ � F 1 · F 2 � σ might NOT be � F 1 � σ · � F 2 � σ 15

A σ , A and Hom + For any φ ∈ Hom + ( A , R ) and any a ∈ A σ φ ( � a · a � σ ) ≥ 0 . Lemma (Razborov) Cauchy-Schwarz inequality for all φ ∈ Hom + ( A , R ) φ ( � a 2 � σ · � b 2 � σ ) ≥ φ ( � ab � 2 σ ) . Special cases: φ ( � a 2 � σ ) ≥ φ ( � a � 2 σ ) φ ( � a 2 � ) ≥ φ ( � a � 2 ) φ ( � σ � σ ) Uses special probability distribution on Hom + ( A σ , R ) related to φ . 16

Let σ and Hom + ( A , R ) be fixed. This gives Skip this. Hard to follow. G 1 , G 2 , G 3 , . . . To make φ σ ∈ Hom + ( A σ , R ), we need a sequence of labeled graphs. Incorrect: Take a copy of σ in each G i and get a convergent G σ 1 , G σ 2 , G σ 3 , . . . Correct: Fix G i . Randomly label a copy of σ and get G σ i . This gives P ( F , G σ n ) for all F ∈ F σ . By a random σ , we get a probability distribution P σ G n on the functions P ( ., G σ n ). These P σ G n then weakly converge to a (unique) probability φ on φ σ ∈ Hom + ( A σ , R ). distribution P σ A crucial feature : if a ∈ A σ and φ ∈ Hom + ( A , R ), then φ [ φ σ ( a )] = φ ( � a � σ ) . φ ( � σ � σ ) · E P σ This can be viewed as an analogue of P ( B ) · P ( A | B ) = P ( A ∧ B ). 17

Skip this. Hard to follow. A crucial feature : if a ∈ A σ and φ ∈ Hom + ( A , R ), then φ [ φ σ ( a )] = φ ( � a � σ ) . φ ( � σ � σ ) · E P σ (1) Bonus: For a given φ ∈ Hom + ( A , R ), exists a unique probability distribution P σ φ satisfying (1). (1) is especially useful when φ σ ( a ) ≥ 0 with probability one. It gives φ ( � a � σ ) ≥ 0. In particular, for any φ ∈ Hom + ( A , R ) and any a ∈ A σ φ ( � a · a � σ ) ≥ 0 . Simplified notation is just � a · a � σ ≥ 0. 18

Skip this. Hard to follow. Consider sequence ( K n , n + K 1 ) n with φ ∈ Hom + ( A , R ) and σ = 1. 1 1 � � There exists φ ′ ∈ Hom + ( A σ , R ) such that φ ′ = 1. 1 But � � � � � � �� = 1 φ σ φ = φ ( � σ � σ ) · E P σ 2 φ 1 1 σ Flag algebras do not see ’rare vertices’. 19

Mantel’s Theorem again � a � 2 1 ≤ � a 2 � 1 For all φ ∈ Hom + ( A , R ) σ is K 1 denoted by 1. Over triangle-free graphs i.e. φ ( K 3 ) = 0 � � 2 � � � � 2 � � ≤ φ φ 1 1 1 1 20

Mantel’s Theorem again � a � 2 1 ≤ � a 2 � 1 For all φ ∈ Hom + ( A , R ) σ is K 1 denoted by 1. Over triangle-free graphs i.e. φ ( K 3 ) = 0 � � 2 � � � � 2 � � � 2 � � � � � ≤ φ φ = φ = φ 1 1 1 1 1 1 20

Mantel’s Theorem again � a � 2 1 ≤ � a 2 � 1 For all φ ∈ Hom + ( A , R ) σ is K 1 denoted by 1. Over triangle-free graphs i.e. φ ( K 3 ) = 0 � � 2 � � � � 2 � � � 2 � � � � � ≤ φ φ = φ = φ 1 1 1 1 1 1 � 2 � 1 � � φ ≤ φ 3 20

Mantel’s Theorem again � a � 2 1 ≤ � a 2 � 1 For all φ ∈ Hom + ( A , R ) σ is K 1 denoted by 1. Over triangle-free graphs i.e. φ ( K 3 ) = 0 � � 2 � � � � 2 � � � 2 � � � � � ≤ φ φ = φ = φ 1 1 1 1 1 1 � 2 � 1 � � φ ≤ φ 3 � 1 + 2 � � � 3 · 3 · φ = φ 20