Characterizing extremal limits

Transcription

1 Characterizing extremal limits Oleg Pikhurko University of Warwick ICERM, 11 February 2015

2 Rademacher Problem g(n, m) := min{#k 3 (G) : v(g) = n, e(g) = m} Mantel 1906, Turán 41: max{m : g(n, m) = 0} = n2 4 Rademacher 41: g(n, n ) = n 2

3 Just Above the Turán Function Erdős 55: m n Erdős 62: m n2 + εn 4 Erdős 55: Is g(n, n2 + q) = q n for q < n/2? 4 2 Kk,k + q edges versus K k+1,k 1 + (q + 1) edges Lovász-Simonovits 75: Yes Lovász-Simonovits 83: m n2 4 + εn2

4 Asymptotic Version g(a) := lim n g(n,a( n 2)) ( n 3) Upper bound: K cn,...,cn,(1 tc)n Moon-Moser 62,Nordhaus-Stewart 62 (Goodman 59): g(a) 2a 2 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 H a n : modify the last two parts of K cn,...,cn,(1 tc)n P.-Razborov 15: almost extremal G n is o(n 2 )-close to some Hn a

5 Possible Edge/Triangle Densities K Upper bound: Kruskal 63, Katona 66

6 Limit Object Subgraph density p(f, G) = Prob { G[ random v(f)-set ] = F } F 0 = {finite graphs} (G n ) converges if v(g n ) and F F 0 lim n p(f, G n ) =: φ(f) LIM = {all such φ} [0, 1] F 0 g(a) = inf{φ(k 3 ) : φ LIM, φ(k 2 ) = a}

7 Razborov s Flag Algebra A 0 φ LIM [0, 1] F 0 F 0 = {unlabeled graphs} RF 0 := {quantum graphs} = { α i F i } Linearity: φ : RF 0 R A 0 := RF 0 / linear relations that always hold φ(f 1 )φ(f 2 ) = c H φ(h) Define: F 1 F 2 := H c HH φ : A 0 R is algebra homomorphism

8 Positive Homomorphisms φ Hom(A 0, R) is positive if F F 0 φ(f) 0 Hom + (A 0, R) = {positive homomorphisms} Lovász-Szegedy 06, Razborov 07: LIM = Hom + (A 0, R) : Let φ Hom + (A 0, R) F =n φ(f) = 1 Distribution on F 0 n Prob[ random Gn φ ] = 1 φ LIM Write α i F i 0 if φ Hom + (A 0, R) α i φ(f i ) 0 Equivalently: (Gn ) lim inf α i p(f i, G n ) 0

9 Limit version of the problem g(a) = min{φ(k 3 ) : φ Hom + (A 0, R), φ(k 2 ) = a} Characterise all extremal φ: {φ Hom + (A 0, R) : φ(k 3 ) = g(φ(k 2 ))} =? Alternatively: work with graphons. E.g. g(a) = min{t(k 3, W ) : graphon W with t(k 2, W ) = a}

10 Razborov s Proof for a [ 1 2, 2 3 ] h(a) = conjectured value Hom + (A 0, R) [0, 1] F is closed f (φ) := φ(k 3 ) h(φ(k 2 )) is continuous φ 0 that minimises f on {φ Hom + (A 0, R) : 1 2 φ(k 2) 2 3 } a := φ 0 (K 2 ) c : e(k cn,cn,(1 2c)n ) a ( ) n 2 compact

11 Goodman bound Goodman bound: K 3 K 2 (2K 2 1) 0 Cauchy-Schwarz: [[K 1 2 K 1 2 ]] 1 K 2 K 2 [[K 1 2 K 1 2 ]] 1 = 1 2 K K P K K 2 Assume 1 < a < Otherwise done by the Goodman bound h is differentiable at a

12 At Most cn Triangles per Edge Pick G n φ 0 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 φ E 0 (K E 3 ) c a.s. Informal explanation: φ E 0 : Two random adjacent roots x 1, x 2 in G n K E 3 : Density of rooted triangles

13 Flag Algebra A E E := (K 2, 2 roots) F E := {(F, x 1, x 2 ) : F F 0, x 1 x 2 } p(f, G): root-preserving induced density G n F E converges if F F E p E (F, G n ) φ E (F) φ E : RF E R A E := ( RF E / trivial relations, multiplication ) Razborov 07: {limits φ E } = Hom + (A E, R) Random homomorphism φ E 0 (K E 3 ): Gn φ (Gn, [random x 1 x 2 ]) M(F E ) Weak limit

14 Vertex Removal Remove x V (G n ): p(k2, G n ) : /( Remove edges: d(x) n ) ( 2 Remove isolated x: n n 1 ) 2)/( 2 = n +... Total change: K2 1(x)/( ) n 2 + a 2 n +... p(k3, G n ) = K3 1(x)/( n 3) + φ0 (K 3 ) 3 n +... Expect: p(k 3 ) h (a) p(k 2 ) Cloning x: signs change Approximate equality for almost all x Flag algebra statement: 3! φ 1 0 (K 1 3 ) + 3φ 0 (K 3 ) = 3c ( 2φ 1 0 (K 1 2 ) + 2a ) a.s.

15 Finishing line Recall: A.s. 3! φ 1 0 (K3 1) + 3φ 0(K 3 ) = 3c ( 2φ 1 0 (K 2 1) + 2a) φ E 0 (K3 E) c Average? 0 = 0 Slack Multiply by K 1 2 & PE 3 and then average! Calculations give φ 0 (K 3 ) 3ac(2a 1) + φ 0(K 4 ) φ 0(K 1,3 ) 3c + 3a 2 φ 0 (K 4 ) 0 & φ 0 (K 1,3 ) 0 φ 0 (K 3 ) h(a)

16 Extremal Limits Extremal limit: limits of almost extremal graphs Equivalently: { φ Hom + (A 0, R) : φ(k 3 ) = g(φ(k 2 )) } P.-Razborov 15: {extremal limits}={limits of Hn s} a Implies the discrete theorem Pick a counterexample (Gn ) Subsequence convergent to some φ H a n φ δ (G n, Hn a ) 0 Overlay V (Gn ) = V (Hn a ) = V 1 V t 1 U G[Vi, V i ] almost complete G[Vi ] almost empty G[U] has o(n 3 ) triangles

17 Structure of Extremal φ 0 Assume φ 0 (K 3 ) = h(a) (= g(a)) with 1 a If a { 1, 2}: 2 3 Goodman s bound is sharp φ0 ( P 3 ) = 0 Complete partite Cauchy-Schwarz regular φ0 is the balanced k-partite limit, done! Suppose a ( 1, 2) 2 3 Density of K 4 and K 1,3 is 0 If φ 0 (P 3 ) = 0, Complete partite K4 -free at most 3 parts done!

18 Case 2: φ 0 (P 3 ) > 0 Special graphs F 1 and F 2 : Claim: φ 0 (F 1 ) = φ 0 (F 2 ) = 0 Claim: Exist many P 3 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 F 1, F 2, or K 1,3 G n [A, B] is almost complete K 4 -freeness + calculations

19 Clique Minimisation Problem Open: Exact result for K 3 Nikiforov 11: Asymptotic solution for K 4 Reiher 15: Asymptotic solution for K r Open: Structure & exact result

20 General Graphs Colour critical: χ(f) = r + 1 & χ(f e) = r Simonovits 68: ex(n, F) = ex(n, Kr+1 ), n n 0 Mubayi 10: Asymptotic for m ex(n, F) + εf n P.-Yilma 15: Asymptotic for m ex(n, F) + o(n 2 ) Bipartite F Conjecture (Erdős-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

21 Thank you!