Negligible Obstructions & Fa r a h Exponents ¥ TEilin Jiang ¥zF¥E¥q Masachusetts Institute of Technology Joint work with I I I I I T a o Jiang I I Xi, J i e M a
TuriinNumber m a x number of edges i n e x (n. F) a n n - ve r te x graph = 1 subgraph. n o F contains a s a # forbidden subgraph CONI F o r every bipartite graph F , t h e r e Q r e x i s t s E s t = t o (N). e x ( n , F ) [ E r d Es 1988]
C O N I (Rational Exponents).tt bipartite F e x ( n . F ) = @ (nr), 7- r e Q : classicalre t.E.I suttsekat. = Off-st). e x (n. Ks, t ) E# [KE vari-Sos-Taran] = S L ( r i t ) e x (n. K a t ) [Kollar-REnyai-Szabo) w h e n t > t o l d . t O (n'+ t ) e x (n. O s t ) [Faudree-Simonovits), = • € § € }y¥ O s t . F = (n't's). e x ( n . Os, t ) • my r [Conlon = w h e n t o ( E ) . t > ← s → @ (n?) OpenProblemse e x (n. Kane) = "beyond o u r reach" @ (n?) ( n . 04.2) e x =
CONI (Realizability) Are Q. n l . 2) 7- bipartite F . . ④ (nr). e x (n. F ) = Breakthrough (Butch. Conlon 20151. # Q n e x ( n . F ) @ ( n r ) F ( l . z ) , F : = * a finite family of forbidden graphs rooted graph graph F equipped RCF) E V CF) D E I A F i s w i t h a 4 = "FY#of n o n - r o o t s . T h e density of F i f f r o o t s e t . yo-1h power of F : disjoint copies of F . FP T h e p = identified a t r o o t s . 1.11 1 × 1*11 E t → f F3 To o k 3 1
CONI (Realizability) free.nl. 2) 7- bipartite F . . ④ (nr), e x (n. F ) = [Bath-Conlon). T H I f F "balanced "rooted-tree F , 7- pent: e x (n. F P ) - n (n2-FF), e.cn I.i nni e a rooted graph. also t r e e . a a e in 4 . is). 7- "balanced"rooted t r e e F . w i t h density p i CONI Hp E Q O ( n ' ¥ ) . Bc-density) e x (n. F P ) = Hp c - a t f (p i s a + H "balanced" rooted F p c - I N TheBakh-Conlonci F . t r e e 0 ( n 2 - ¥ ) . e x ( n . F P ) = 2 conjectures. "balanced" condition necessary i n t h e above 12M£ i s
⇒ in 4 . is). 7- "balanced"rooted t r e e F . w i t h density p It P CONI E Q i 2 - s t ) . Bc-density) e x (n. 0 ( n F P ) = Hp c - I N + l p i s a EE., goin knowing g ¥÷÷:" 11¥. i.FI?...F. ± . * - ¥¥¥¥¥ ← s o I f s on a a go o o ¥I t f - f , g s . = = "FY.tt of n o n - r o o t s . / l Kang- K i m - L i u , Erda's-Simonov its]. L E I f F DC density. t h t t c- IN). I f s o i s 8 t p i s m a {Jiang i tha. Yepreunyan. Kang, K i m , L i u . Conlon, Ta n z e r, L e e . Qiu}. b Yorio [Jiang-J.-Ma]. T H I a s µ ,¥, density. B c 1 then f 1%13 s = I > l , i f i s H p t a a t fo. (on?2) t ¥ , m New B o densities: - M
[Jiang-J.-Ma): T H I s µ ,¥, density. B c 1 then f = be> 1. i f 1%13 s i s H p t a a # N . j ⇐ ¥¥¥¥¥I Ts . t t ' H s , t EIN? s , s'-I. s ' E IN. T i m w i t h t O ( n' ¥ ) e x (n. FP) = if F. = Ts. t . s i i s balanced, t h e n framework & Application
f ±÷÷÷÷:::::::::........... E E F o - o - o - o . = ← 2 - YPF) for all p . I f 4/3. Goal: e x ( n . FP I 0 ( n = = w (n'-YPF). degree d of A ② = {embeddings from F t o G } Find p ample Find FP i n G . RMI Goal KEG embedding { F o r a } . embedding y from = : D E I A n t o G F i s ↳ G } # { F a n injection 4 : V l a V ( F l s e t . → F - subgraph counts. i n G . I i n £ . U z . i n F . 9 ( U r l H U , Y l a , I ~ n w (n't t'-'delete') I N E ' l, ( n delF') w ( n { F e a t } O B I * = h = = e n . i FOG An embedding n i s C- ample if a t . they I . D E I 4...... Me a r e a r e pair w i s e disjoint R t e ) , but images of non-roots identical o n .
↳ E - - O ÷ ⇐ ÷÷ ÷÷÷÷÷ * . w ( n I M F ' l) # { F G } O B I : o r = K G ) : # { F G I o } w ( L ) 7- R ( F ) = T : → 'to embeddings from F "agreeing" w i t h ← £ . ⇐ ÷÷÷÷ Ideally € 1 / → pp @ F Images of pair w i s e disjoint. n o n - r o o t s a r e
Possiblewaystogowrongn i÷ ⇐ ÷÷o ⇐ ÷÷o (none of - a ) ← a which i s a n obstruction family for F A family FE of subtrees of F D E F i s 0 . after adding U if € {non-roots 3. tf U U t t o w i t h t h e resulting rooted graph c o n t a i n s RLF), r o o t s e t t h e Fo ⇐ F ERE).). (Fo member of Fo subgraph. rooted a s a a R I F o ) E E = {• }. F o f n o - o n e , • → → • → → → → = obstruction family for F E
⇒ ⇒ ↳ Additionalassumption obstruction family Fo for F . Give a n e l El). # {Fo walls G} H Fo E F o 0 ( n d = will-ample embeddings from Fo t o A s a } I U those Y : F t h a t "extends" G { F Consider I i w 111-ample Mo: FIG}. a = f o eff a n f o & 7- Y': Fo ↳ F s . K F I n § ° £ A {extension of f o ¥ o ( n d ecfo7.de#I-eeFo)) G } = # o ( n d ece'). = Find EP in G w ( n IRCF") w (nd-CFI) l I I = = IT l t } - w ( 1 ) v ( k ) : # { I Cannot go wrong. R E I I → o . . . .
UpshotofthoughtEoperimentn w ( n ' t FF). ② degree of = A i s regular graph. ① . "additional assumption" o n F o (obstruction family for F ) ③ c a n find F P i n G . T h e n ETO i s negligible for F , i f Given Fo and F . w e say Fo DEI o . I c o > O and M E I N E Nt. F P E > s - t , degrees of a - n - v e r t e x graph G- i f f and a r e c > c o 5 " K e n t ( x - i - i f f ) c . nd, a n d b e t w e e n E (eton) indeed # { Fo Mus G ) and { F I s G 3=0. t h e n . Given Fo. If every member of F o i s Item (Negligibility lemma). n e g . for F 2 - 'GF) e - ( n . F P) p E N T 0 ( n f t h e n =
Framework & Application Tse.tt#..IIfII..IT; oooo sea ooo Consider S ' - I . 5 = 2 . Obstruction family: ° ¥ , F - All A l l 's { d in . n o n ' s @ @ @ • • • i f t I
l Kang- K i m - L i u , E r di's-Simonov its]. L E I BC density. t h t t c- IN). I f s o i s 8 t p i s m a t § Y s . a c- I N t : i s BC-density. I CONDI: 1 N . t s e a . m M NJ M X X f Eft I ma EE see
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