Characterizing extremal limits Oleg Pikhurko University of Warwick - - PowerPoint PPT Presentation

Characterizing extremal limits

Oleg Pikhurko

University of Warwick

ICERM, 11 February 2015

Rademacher Problem

◮ g(n, m) := min{#K3(G) : v(G) = n, e(G) = m} ◮ Mantel 1906, Turán’41: max{m : g(n, m) = 0} = ⌊ n2 4 ⌋ ◮ Rademacher’41: g(n, ⌊ n2 4 ⌋ + 1) = ⌊ n 2⌋

Just Above the Turán Function

◮ Erd˝

• s’55: m ≤ ⌊ n2

4 ⌋ + 3 ◮ Erd˝

• s’62: m ≤ ⌊ n2

4 ⌋ + εn ◮ Erd˝

• s’55: Is g(n, ⌊ n2

4 ⌋ + q) = q · ⌊ n 2⌋ for q < n/2 ?

◮ Kk,k + q edges versus Kk+1,k−1 + (q + 1) edges

◮ Lovász-Simonovits’75: Yes ◮ Lovász-Simonovits’83: m ≤ ⌊ n2 4 ⌋ + εn2

Asymptotic Version

◮ g(a) := limn→∞ g(n,a(n

2))

(n

3)

◮ Upper bound: Kcn,...,cn,(1−tc)n ◮ Moon-Moser’62,Nordhaus-Stewart’62 (Goodman’59):

g(a) ≥ 2a2 − a

◮ Bollobás’76: better lower bound ◮ Fisher’89: g(a) for 1 2 ≤ a ≤ 2 3 ◮ Razborov’08: g(a) for all a ◮ No stability

◮ Ha

n: modify the last two parts of Kcn,...,cn,(1−tc)n ◮ P

.-Razborov ≥’15: ∀ almost extremal Gn is o(n2)-close to some Ha

n

Possible Edge/Triangle Densities

0.2 0.4 0.6 0.8 1.0 K2 0.2 0.4 0.6 0.8 1.0 K3 0.2 0.4 0.6 0.8 1.0 K2 0.2 0.4 0.6 0.8 1.0 K3

◮ Upper bound: Kruskal’63, Katona’66

Limit Object

◮ Subgraph density

p(F, G) = Prob

• G[ random v(F)-set ] ∼

= F

• ◮ F0 = {finite graphs}

◮ (Gn) converges if v(Gn) → ∞ and

∀ F ∈ F0 ∃ lim

n→∞ p(F, Gn) =: φ(F) ◮ LIM = {all such φ} ⊆ [0, 1]F0 ◮ g(a) = inf{φ(K3) : φ ∈ LIM, φ(K2) = a}

Razborov’s Flag Algebra A0

◮ φ ∈ LIM ⊆ [0, 1]F0 ◮ F0 = {unlabeled graphs} ◮ RF0 := {quantum graphs} = { αiFi} ◮ Linearity: φ : RF0 → R ◮ A0 := RF0/

• linear relations that always hold
• ◮ φ(F1)φ(F2) = cHφ(H)

◮ Define: F1 · F2 := H cHH ◮ φ : A0 → R is algebra homomorphism

Positive Homomorphisms

◮ φ ∈ Hom(A0, R) is positive if ∀ F ∈ F0 φ(F) ≥ 0 ◮ Hom+(A0, R) = {positive homomorphisms} ◮ Lovász-Szegedy’06, Razborov’07:

LIM = Hom+(A0, R)

◮ ⊇: Let φ ∈ Hom+(A0, R)

◮

|F|=n φ(F) = 1

◮ Distribution on F0

n

◮ Prob[ random Gn → φ ] = 1 ◮ φ ∈ LIM

◮ Write αiFi ≥ 0 if

◮ ∀φ ∈ Hom+(A0, R) αiφ(Fi) ≥ 0 ◮ Equivalently: ∀(Gn)

lim inf αip(Fi, Gn) ≥ 0

Limit version of the problem

◮ g(a) = min{φ(K3) : φ ∈ Hom+(A0, R), φ(K2) = a} ◮ Characterise all extremal φ:

{φ ∈ Hom+(A0, R) : φ(K3) = g(φ(K2))} = ?

◮ Alternatively: work with graphons. E.g.

g(a) = min{t(K3, W) : graphon W with t(K2, W) = a}

Razborov’s Proof for a ∈ [1

2, 2 3]

◮ h(a) = conjectured value ◮ Hom+(A0, R) ⊆ [0, 1]F is closed

⇒ compact

◮ f(φ) := φ(K3) − h(φ(K2)) is continuous ◮ ∃ φ0 that minimises f on

{φ ∈ Hom+(A0, R) : 1

2 ≤ φ(K2) ≤ 2 3} ◮ a := φ0(K2) ◮ c : e(Kcn,cn,(1−2c)n) ≈ a

n

2

Goodman bound

◮ Goodman bound: K3 − K2(2K2 − 1) ≥ 0

◮ Cauchy-Schwarz: [

[K 1

2 · K 1 2 ]

]1 ≥ K2 · K2

◮ [

[K 1

2 · K 1 2 ]

]1 = 1

2 K3 + 1 2 K2 − 1 6 ¯

P3 ≤ 1

2 K3 + 1 2 K2 ◮ Assume 1 2 < a < 2 3

◮ Otherwise done by the Goodman bound

◮ h is differentiable at a

At Most cn Triangles per Edge

◮ Pick Gn → φ0

◮ Rate of growth: g(n, m + 1) − g(n, m) ≈ cn ◮ ≈ cn triangles per new edge ◮ Gn has cn triangles on almost every edge

◮ Flag algebra statement

φE

0 (K E 3 ) ≤ c

a.s.

◮ Informal explanation:

◮ φE

0 : Two random adjacent roots x1, x2 in Gn

◮ K E

3 : Density of rooted triangles

Flag Algebra AE

◮ E := (K2, 2 roots) ◮ FE := {(F, x1, x2) : F ∈ F0, x1 ∼ x2} ◮ p(F, G): root-preserving induced density ◮ Gn ∈ FE converges if ∀F ∈ FE pE(F, Gn) → φE(F) ◮ φE : RFE → R ◮ AE :=

• RFE/trivial relations, multiplication
• ◮ Razborov’07: {limits φE} = Hom+(AE, R)

◮ Random homomorphism φE 0 (K E 3 ):

◮ Gn → φ ◮ (Gn, [random x1 ∼ x2]) ∈ M(FE) ◮ Weak limit

Vertex Removal

◮ Remove x ∈ V(Gn):

◮ ∂ p(K2, Gn) : ◮ Remove edges: −d(x)

n

2

• ◮ Remove isolated x: ×

n

2

n−1

2

• = 1 + 2

n + ...

◮ Total change: −K 1

2 (x)/

n

2

• + a 2

n + ...

◮ ∂ p(K3, Gn) = −K 1

3 (x)/

n

3

• + φ0(K3) 3

n + ... ◮ Expect: ∂p(K3) h′(a) ∂p(K2) ◮ Cloning x: signs change ◮ Approximate equality for almost all x ◮ Flag algebra statement:

−3! φ1

0(K 1 3 ) + 3φ0(K3) = 3c

• −2φ1

0(K 1 2 ) + 2a

• a.s.

Finishing line

◮ Recall: A.s.

◮ −3! φ1

0(K 1 3 ) + 3φ0(K3) = 3c

• −2φ1

0(K 1 2 ) + 2a

• ◮ φE

0 (K E 3 ) ≤ c ◮ Average?

◮ 0 = 0 ◮ Slack

◮ Multiply by K 1 2 & P E 3 and then average! ◮ Calculations give

φ0(K3) ≥ 3ac(2a − 1) + φ0(K4) + 1

4φ0(K 1,3)

3c + 3a − 2

◮ φ0(K4) ≥ 0 & φ0(K 1,3) ≥ 0

⇒ φ0(K3) ≥ h(a)

Extremal Limits

◮ Extremal limit: limits of almost extremal graphs ◮ Equivalently: { φ ∈ Hom+(A0, R) : φ(K3) = g(φ(K2)) } ◮ P

.-Razborov ≥’15: {extremal limits}={limits of Ha

n’s} ◮ Implies the discrete theorem

◮ Pick a counterexample (Gn) ◮ Subsequence convergent to some φ ◮ Ha

n → φ

◮ δ(Gn, Ha

n) → 0

◮ Overlay V(Gn) = V(Ha

n) = V1 ∪ · · · ∪ Vt−1 ∪ U

◮ G[Vi, Vi] almost complete ◮ G[Vi] almost empty ◮ G[U] has o(n3) triangles

Structure of Extremal φ0

◮ Assume φ0(K3) = h(a) (= g(a)) with 1 2 ≤ a ≤ 2 3 ◮ If a ∈ {1 2, 2 3}:

◮ Goodman’s bound is sharp ◮ φ0(¯

P3) = 0

◮ Complete partite ◮ Cauchy-Schwarz

⇒ regular

◮ φ0 is the balanced k-partite limit, done!

◮ Suppose a ∈ ( 1 2, 2 3) ◮ Density of K4 and K 1,3 is 0 ◮ If φ0(P3) = 0,

◮ Complete partite ◮ K4-free ⇒ at most 3 parts ⇒ done!

Case 2: φ0(P3) > 0

◮ Special graphs F1 and F2: ◮ Claim: φ0(F1) = φ0(F2) = 0 ◮ Claim: Exist many P3’s st

◮ |A| = Ω(n): vertices sending 3 edges to it ◮ |B| = Ω(n): vertices sending ≤ 2 edges to it

◮ Non-edge across → a copy of F1, F2, or K 1,3 ◮ Gn[A, B] is almost complete ◮ K4-freeness + calculations

⇒

Clique Minimisation Problem

◮ Open: Exact result for K3 ◮ Nikiforov’11: Asymptotic solution for K4 ◮ Reiher ≥’15: Asymptotic solution for Kr ◮ Open: Structure & exact result

General Graphs

◮ Colour critical: χ(F) = r + 1 & χ(F − e) = r

◮ Simonovits’68: ex(n, F) = ex(n, Kr+1), n ≥ n0 ◮ Mubayi’10: Asymptotic for m ≤ ex(n, F) + εFn ◮ P

.-Yilma ≥’15: Asymptotic for m ≤ ex(n, F) + o(n2)

◮ Bipartite F

◮ Conjecture (Erd˝

• s-Simonovits’82, Sidorenko’93):

◮ Random graphs are optimal ◮ ..., Conlon-Fox-Sudakov’10, Li-Szegedy ≥’15,

Kim-Lee-Lee ≥’15, ...

◮ Forcing Conjecture (Li-Szegedy): F biparite non-tree

⇒ extremal graphons are constants

Thank you!