Elusive problems in extremal graph theory Andrzej Grzesik (Krakow) - PowerPoint PPT Presentation

Elusive problems in extremal graph theory Andrzej Grzesik (Krakow) Dan Kr´ al’ (Warwick) L´ aszl´ o Mikl´ os Lov´ asz (MIT/Stanford) Monash 24/4/2017 1

Tur´ an Problems • Maximum edge-density of H -free graph 1 • Mantel’s Theorem (1907): 2 for H = K 3 ( K n 2 ) 2 , n ℓ − 2 • Tur´ an’s Theorem (1941): ℓ − 1 for H = K ℓ ( K ℓ − 1 ) n n ℓ − 1 ,..., χ ( H ) − 2 • Erd˝ os-Stone Theorem (1946): χ ( H ) − 1 • extremal examples unique up to o ( n 2 ) edges 2

Edge vs. Triangle Problem • Minimum density of K 3 for a specific edge-density • determined by Razborov (2008), K α n,..., α n, (1 − k α ) n • extensions by Nikiforov (2011) and Reiher (2016) for K ℓ • Pikhurko and Razborov (2017) gave extremal examples generally not unique, can be made unique by K n = 0 3

Another example • Minimum sum of densities of K 3 and K 3 • Goodman’s Bound (1959): K 3 + K 3 ≥ 1 4 every n/ 2-regular graph is a minimizer • minimizer can be made unique K 3 = 0, or K 3 = 0, or C 4 = 1 / 16 (Erd˝ os-R´ enyi random graph G n, 1 / 2 ) 4

This Talk • Conjecture (Lov´ asz 2008, Lov´ asz and Szegedy 2011) Every finite feasible set H i = d i , i = 1 , . . . , 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., Lov´ asz Jr.): FALSE 5

Limits of dense graphs • 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 α n,n , blow ups G [ K n ] Erd˝ os-R´ enyi random graphs G n,p , planar graphs • graphon W : [0 , 1] 2 → [0 , 1], s.t. W ( x, y ) = W ( y, x ) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 6

Finitely forcible graph limits • a graphon W is finitely forcible if there exist H 1 , . . . , H k and d 1 , . . . , d k such that W is the only graphon with the expected density of H i equal to d i • ⇔ the only graphon minimizing � α j d ( H ′ j , W ) density calculation: � ( H i − d i ) 2 = � α j H ′ j • Lov´ asz, S´ os (2008): Step graphons are finitely forcible. 7

Statement of Problem • Conjecture (Lov´ asz 2008, Lov´ asz and Szegedy 2011): Every extremal problem min � α j d ( H j , W ) has a finitely forcible optimal solution. • extremal graph theory problem → finitely forcible optimal solution → “simple structure” gives new bounds on old problems • Conjectures (Lov´ asz and Szegedy): The space T ( W ) of a finitely forcible W is compact. The space T ( W ) has finite dimension. 8

Finitely forcible graph limits • Theorem (Cooper, K., Martins): Every graphon is a subgraphon of a finitely forcible graphon. Q A B C D E F G P R A B C D E F W F G P Q R 9

Main result • Theorem (Grzesik, K., Lov´ asz Jr.) ∃ graphon family W , graphs H i , reals d i , i = 1 , . . . , m W ∈ W ⇔ d ( H i , W ) = d i for i = 1 , . . . , m no graphon in W is finitely forcible A B C D A D B D C D D D E D F D G E F G H A B C D A D B D C D D D E D F D G E F G H 10

Some details of the proof z ∈ [0 , 1] N • graphons W P ( ⃗ z ), ⃗ z satisfies polynomial inequalities in P (e.g. z 1 + z 2 2 ≤ 1) ⃗ • construct J i ⊆ [0 , 1], inequalities P and inj. maps f i f i ( x 1 , . . . , x i ) = ( z 1 , . . . , z ( i +1)( i +2) ) 2 ( x 1 ) → ( z 1 , z 2 ), ( x 1 , x 2 ) → ( z 1 , z 2 , z 3 , z 4 , z 5 ), etc. d ( H i , W P ( ⃗ z )) = g i ( x 1 , . . . , x i ) if x k ∈ J k , k ∈ N each J i has positive measure • ⇒ no graphon in W P ( ⃗ z ) is finitely forcible 11

Some details of the proof z ∈ [0 , 1] N • graphons W P ( ⃗ z ), ⃗ z satisfies polynomial inequalities in P (e.g. z 1 + z 2 2 ≤ 1) ⃗ • independent of P : there exist graphs H 1 , . . . , H k there exist polynomials q 1 , . . . , q ℓ in d ( H i , W ) • for every P : there exist reals α 1 , . . . , 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 maps f i by polynomial inequalities 12

Possible extensions • techniques universal to prove more general results equalize other functions than subgraph densities • Theorem (Grzesik, K., Lov´ asz Jr.) ∃ graphon family W , graphs H i , reals d i , i = 1 , . . . , m W ∈ W ⇔ d ( H i , W ) = d i for i = 1 , . . . , m no graphon in W is finitely forcible all graphons in W have the same entropy • extremal problems with no typical structure 13

Thank you for your attention! 14