Set Systems and Families of Permutations with Small Traces
We study the maximum size of a set system on $n$ elements whose trace on any $b$ elements has size at most $k$. We show that if for some $b \ge i \ge 0$ the shatter function $f_R$ of a set system $([n],R)$ satisfies $f_R(b) < 2^i(b-i+1)$ then $|R| = …
Authors: Otfried Cheong (KAIST), Xavier Goaoc (INRIA Lorraine - LORIA), Cyril Nicaud (IGM)
apport de recherche ISSN 0249-6399 ISRN INRIA/RR--7154--FR+ENG INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Set Systems and F amilies of P ermutations with Small T races Otfried Cheong — Xavier Goaoc — Cyril Nicaud N° 7154 December 2009 Centre de recherche INRIA Nancy – Grand Est LORIA, T echnop ôle de Nancy-Brabois, Campus scientifique, 615, rue du Jardin Botaniqu e, BP 101, 54602 V illers-Lès-Nancy Téléphone : +33 3 83 59 30 00 — Télécopie : +33 3 83 27 83 19 Set Systems and F amilies of P erm utations with Small T raes Otfried Cheong ∗ , Xa vier Goao † , Cyril Niaud ‡ Thème : Algorithmique, alul ertié et ryptographie Équip e-Pro jet Végas Rapp ort de re her he n ° 7154 Deem b er 2009 11 pages Abstrat: W e study the maxim um size of a set system on n elemen ts whose trae on an y b elemen ts has size at most k . W e sho w that if for some b ≥ i ≥ 0 the shatter funtion f R of a set system ([ n ] , R ) satises f R ( b ) < 2 i ( b − i + 1) then | R | = O ( n i ) ; this generalizes Sauer's Lemma on the size of set systems with b ounded V C-dimension. W e use this b ound to delineate the main gro wth rates for the same problem on families of p erm utations, where the trae orresp onds to the inlusion for p erm utations. This is related to a question of Raz on families of p erm utations with b ounded V C-dimension that generalizes the Stanley-Wilf onjeture on p erm utations with exluded patterns. Key-w ords: Set systems, V C dimension, Sauer Lemma, P erm utation pattern Otfried Cheong w as supp orted b y the K orea Siene and Engineering F oundation Gran t R01-2008-000-11607-0 funded b y the K orea go v ernmen t. The ollab oration b et w een Otfried Cheong and Xa vier Goao w as supp orted b y the INRIA Équip e asso ié e KI. ∗ Theory of Computation Lab, Dept. of Computer Siene, KAIST, K orea. otfriedkaist.edu † Loria, INRIA Nany Grand Est, F rane. goaoloria.fr ‡ LIGM Univ ersité P aris Est, F rane. niauduniv-mlv.fr Set Systems and F amilies of P erm utations with Small T raes Résumé : Nous étudions la taille maximale d'un h yp ergraphe à n sommets don t la trae sur toute sous-famille de b sommets est de taille au plus k . Nous mon trons que p our tous en tiers b ≥ i ≥ 0 , si la fontion de pulv érisation f R d'un h yp ergraphe ([ n ] , R ) satisfait f R ( b ) < 2 i ( b − i + 1) alors | R | = O ( n i ) ; ela généralise le Lemme de Sauer sur la taille des h yp ergraphes de dimension de V apnik-Cherv onenkis b ornée. Nous utilisons ensuite ette b orne p our séparer les prinipaux régimes de roissane p our une question analogue sur les familles de p erm utations, où l'op ération de trae orrespnd à l'inlusion de motifs. Cela est relié à une question de Raz sur les familles de p erm utations à dimension de V apnik-Cherv onenkis b ornée qui généralise la onjeture de Stanley-Wilf sur les p erm utations à motifs exlus. Mots-lés : Hyp ergraphes, Dimension de V apnik-Cherv onenkis, Lemme de Sauer, Motifs de p erm utation Set Systems and F amilies of Permutations with Smal l T r a es 3 1 In tro dution In this pap er, w e study t w o problems of the follo wing a v or: ho w large an a family of om binatorial ob jets dened on [ n ] = { 1 , . . . , n } b e if its n um b er of distint pro jetions on an y small subset is b ounded? W e onsider set systems, where the pro jetion is the standard notion of trae, and families of p erm uta- tions, where the pro jetion orresp onds to the notion of inlusion used in the study of p erm utations with exluded patterns. Set systems. A set system , also alled a r ange sp a e or a hyp er gr aph , is a pair ( G, R ) where G is a set, the gr ound set , and R is a set of subsets of G , the r anges . Sine w e will only onsider nite set systems, our ground set will alw a ys b e [ n ] . Giv en X ⊂ [ n ] , the tr a e of R on X , denoted R | X , is the set { A ∩ X | A ∈ R } . Giv en an in teger b , let R b denote the set of b -tuples of R , and dene: f R ( b ) = max X ∈ ( [ n ] b ) | R | X | . The funtion f R is alled the shatter funtion of ([ n ] , R ) , and oun ts the size of the largest trae on a subset of [ n ] of size b . The rst problem w e onsider is the follo wing: Question 1. Giv en b and k , ho w large an a set system ([ n ] , R ) b e if f R ( b ) ≤ k ? F or k = 2 b − 1 , the answ er is giv en b y Sauer's Lemma [15 ℄ (also pro v en inde- p enden tly b y P erles and Shelah [17 ℄ and V apnik and Cherv onenkis [18 ℄), whi h states that: | R | ≤ b − 1 X i =0 n i = O ( n b − 1 ) . (1) The largest b su h that f R ( b ) = 2 b is kno wn as the V C-dimension of ([ n ] , R ) . The theory of set systems of b ounded V C-dimension, and in partiular Sauer's Lemma, has man y appliations, in partiular in geometry and appro ximation algorithms; lassial examples inlude the epsilon-net Theorem [7℄ or impro v ed appro ximation algorithms for geometri set o v er [ 6℄. F or the ase of graphs, that is, set systems where all ranges ha v e size 2 , Question 1 is a lassial problem kno wn as a Dir a-typ e pr oblem : what is the maxim um n um b er E x ( n, m, k ) of edges in a graph on n v erties whose indued subgraph on an y m v erties has at most k edges? These problems w ere ex- tensiv ely studied in extremal graph theory sine the 1960's, and w e refer to the surv ey of Griggs et al. [11 ℄ for an o v erview. In the ase of general set sys- tems, the only results w e are a w are of are due to F rankl [9 ℄ and Bollobás and Radlie [5 ℄. Sp eially , F rankl pro v ed that f R (3) ≤ 6 ⇒ | R | ≤ t 2 ( n ) + n + 1 and f R (4) ≤ 10 ⇒ | R | ≤ t 3 ( n ) + n + 1 , where t i ( n ) denotes the n um b er of edges of the T urán graph T i ( n ) . Bollobás and Radlie sho w ed that: f R (4) ≤ 11 ⇒ | R | ≤ n 2 + n + 1 exept for n = 6 . RR n ° 7154 4 Che ong, Go ao & Ni aud There has also b een in terest in the ase where b = αn and b = n − Θ (1) ; w e refer to the artile of Bollobás and Radlie [ 5℄ for an o v erview of these results. P erm utations. The notion of V C-dimension w as extended to sets of p erm u- tations b y Raz [ 14 ℄ as follo ws. Let σ b e a p erm utation on [ n ] and X some subset of [ n ] . The r estrition of σ to X is the p erm utation σ | X of X su h that for an y u, v ∈ X , σ − 1 | X ( u ) < σ − 1 | X ( v ) whenev er σ − 1 ( u ) < σ − 1 ( v ) ; if w e onsider a p erm utation as an ordering, σ | X is simply the order indued on X b y σ . This allo ws to dene the shatter funtion of a set F of p erm utations similarly: φ F ( m ) = max X ∈ ( [ n ] m ) | F | X | . The V C-dimension of F is then the largest m su h that φ F ( m ) = m ! , and the analogue of Question 1 arises naturally for sets of p erm utations: Question 2. Giv en m and k , ho w large an a set F of p erm utations on [ n ] b e if φ F ( m ) ≤ k ? Raz [14℄ sho w ed that an y family of p erm utations on [ n ] su h that φ F (3) < 6 has size at most exp onen tial in n , and ask ed whether the same holds whenev er k < m ! . This problem is related to lassial questions on families of p erm utations with exlude d p attern . A p erm utation σ on [ n ] ontains a p erm utation τ on [ m ] if there exists a 1 < a 2 < . . . < a m in [ n ] su h that σ − 1 ( a i ) < σ − 1 ( a j ) whenev er τ − 1 ( i ) < τ − 1 ( j ) . If no p erm utation in a family F on tains τ then F avoids τ and τ is an exlude d p attern for F . The study of families of p erm utations with exluded patterns go es ba k to a w ork of Kn uth [ 12 ℄, motiv ated b y sorting p erm utations using queues, and reeiv ed onsiderable atten tion o v er the last deades. In partiular, Stanley and Wilf ask ed whether for an y xed p erm uta- tion τ the n um b er of p erm utations on [ n ] that a v oid τ is at most exp onen tial in n , a question answ ered in the p ositiv e b y Marus and T ardos [ 13 ℄. If a family of p erm utations has V C-dimension at most m − 1 then for an y m -tuple X ⊂ [ n ] there is a p erm utation σ ( X ) on [ m ] whi h is forbidden for restritions to X . In that sense, Raz's question generalizes that of Stanley and Wilf. Our results. In this pap er, w e generalize Sauer's Lemma, and sho w that for an y range spae ([ n ] , R ) , if f R ( b ) < 2 i ( b − i + 1) for some b > i ≥ 0 then | R | = O ( n i ) (Theorem 2 ). W e then pro v e that the ondition f R ( b ) = k is in fat e quivalent to a Dira-t yp e problem on graphs for k ≤ 8 + 3 ⌊ b − 3 2 ⌋ + s ( b ) , where s ( b ) = 1 when b is ev en and 0 otherwise (Lemma 3). It follo ws that some onditions f R ( b ) = k lead to gro wth rates with frational exp onen ts (Corol- lary 4), a b eha vior not aptured b y Theorem 2. Finally , w e giv e a redution of the p erm utation problem to the set system problem (Lemma 5 ) from whi h w e dedue the main transitions b et w een the onstan t, p olynomial and at least exp onen tial b eha viors for Question 2. INRIA Set Systems and F amilies of Permutations with Smal l T r a es 5 2 Set systems In this setion w e giv e b ounds on the size of a set R of ranges on [ n ] with a giv en f R ( b ) . Reall that a set system ([ n ] , R ) is ide al , also alled monotone de r e asing 1 , if for an y B ⊂ A ∈ R w e ha v e B ∈ R . The next lemma w as pro v en, indep enden tly , b y Alon [1℄ and F rankl [9 ℄. Lemma 1. F or any set system ([ n ] , R ) ther e exists an ide al set system ([ n ] , ˜ R ) suh that | R | = | ˜ R | and for any inte ger b we have f ˜ R ( b ) ≤ f R ( b ) . This an b e sho wn b y dening, for an y x ∈ [ n ] , the op erator (also alled a push-down or a ompr ession ) ˜ T x ( R ) = { A \ { x } | A ∈ R } ∪ { A | A ∈ R su h that x ∈ A and A \ { x } ∈ R } , that remo v es x from an y range in R where that do es not derease the total n um b er of sets. Then, ˜ R = ˜ T 1 ˜ T 2 . . . ( ˜ T n ( R ) . . . is one su h ideal set. W e refer to Bollobás [ 4 , Chapter 17℄ and the surv ey of Füredi and P a h [10 ℄ for more details. An immediate onsequene of Lemma 1 is that w e an w ork with ideal set systems when studying our rst question. 2.1 Sauer's Lemma for small traes Dene n − 1 = 0 and onsider the sequene υ i ( b ) = 2 i ( b − i + 1) that in terp olates b et w een b + 1 = υ 0 ( b ) and 2 b = υ b − 1 ( b ) . Our rst result is the follo wing generalization of Sauer's Lemma. Theorem 2. L et b > i ≥ 0 b e two inte gers. A ny r ange sp a e ([ n ] , R ) with f R ( b ) < υ i ( b ) has size | R | = f R ( n ) < P i j =0 ( b − j + 1) n j Pr o of. By Bondy's Theorem [4 ℄, for an y b + 1 distint ranges there exist b elemen ts on whi h they ha v e distint trae. It follo ws that if f R ( b ) < b + 1 w e also ha v e f R ( n ) < b + 1 for an y n , and the statemen t holds for i = 0 . Also, from i X j =0 ( b − j + 1 ) b j ≥ ( b − i + 1) i X j =0 b j ≥ ( b − i + 1)2 i = υ i ( b ) , w e ha v e that the statemen t holds for n = b and an y i . No w, w e x b and assume that w e ha v e f R ( b ) < υ k ( b ) ⇒ f R ( t ) < k X j =0 ( b − j + 1) t j whenev er k < i or k = i and t < n . Let R ′ = R | [ n − 1] denote the trae of R on [ n − 1] and let D denote the ranges in R ′ that are the trae of t w o distint ranges from R . Notie that: | R | = | R ′ | + | D | and f R ′ ( b ) < υ i ( b ) . (2) 1 An ideal set system is also an abstr at simpliial omplex to whi h the empt y set w as added. RR n ° 7154 6 Che ong, Go ao & Ni aud Sine D ⊂ R ′ , w e ha v e that | D | X | ≤ | R ′ | X | and th us | D | X | ≤ 1 2 | R | ( X ∪{ n } ) | . It follo ws that f D ( b − 1) ≤ j f R ( b ) 2 k . No w, from υ i ( b ) = 2 υ i − 1 ( b − 1) w e get that: f D ( b − 1) < υ i − 1 ( b − 1) . (3) F rom Equations (2 ) and (3) and the indution h yp othesis w e obtain: | R | < i X j =0 ( b − j + 1) n − 1 j + i − 1 X j =0 ( b − 1 − j + 1) n − 1 j . This rewrites as | R | < b + 1 + i X j =1 ( b − j + 1) n − 1 j + i X j =1 ( b − j + 1) n − 1 j − 1 and with n − 1 j + n − 1 j − 1 = n j w e get | R | < b + 1 + i X j =1 ( b − j + 1) n j = i X j =0 ( b − j + 1) n j , and th us: f R ( b ) < υ i ( b ) ⇒ f R ( n ) < i X j =0 ( b − j + 1) n j . The statemen t follo ws b y indution. No w, onsider the follo wing family of lo w er b ounds. F or i = 1 , . . . , b let λ i ( b ) = max b = b 1 + ...b i i Y j =1 ( b j + 1) and onsider the system ([ n ] , R ) where R is obtained b y splitting [ n ] in to i roughly equal subsets and pi king all i -tuples on taining one elemen t from ea h subset. Notie that | R | = Ω( n i ) and that f R ( b ) ≤ λ i ( b ) . The same holds for i = 0 with λ 0 ( b ) = 1 . Th us, for an y k su h that λ i ( b ) ≤ k < υ i ( b ) , the maxim um size of a set system ([ n ] , R ) with f R ( b ) = k is Θ( n i ) . b υ 0 ( b ) -1 λ 1 ( b ) υ 1 ( b ) -1 λ 2 ( b ) υ 2 ( b ) − 1 λ 3 ( b ) υ 3 ( b ) − 1 λ 4 ( b ) υ 4 ( b ) − 1 λ 5 ( b ) | R | O (1) Ω( n ) O ( n ) Ω( n 2 ) O ( n 2 ) Ω( n 3 ) O ( n 3 ) Ω( n 4 ) O ( n 4 ) Ω( n 5 ) 2 2 3 3 3 4 5 6 4 4 5 7 9 11 12 5 5 6 9 12 15 18 23 24 6 6 7 1 1 16 19 27 31 36 47 48 T able 1: The v alues v i ( b ) and λ i ( b ) for small b . Gaps app ear in red. In partiular, the order of magnitude giv en b y Theorem 2 is tigh t for all b ≤ 4 , with the exeption of set systems with f R (4) = 8 . INRIA Set Systems and F amilies of Permutations with Smal l T r a es 7 Remark. Observ e that the ondition that f R ( b ) < υ i ( b ) do es not imply that R has V C-dimension at most i . A simple example is giv en b y R = { A | A ⊂ [ i ] } ∪ {{ x } | x ∈ [ n ] } , whi h has V C-dimension i and for whi h f R ( b ) = 2 i + b − i − 1 is smaller than υ i − 1 ( b ) = 2 i − 1 ( b − i ) for b large enough. 2.2 Equiv alene with Dira-t yp e problems Reall that E x ( n, m, k ) denotes the maxim um n um b er of edges in a graph on n v erties whose indued subgraph on an y m v erties has at most k edges. Let ζ ( b ) = 8 + 3 ⌊ b − 3 2 ⌋ + s ( b ) where s ( b ) = 1 if b is ev en and 0 otherwise. Lemma 3. F or any b ≥ 3 , the maximal size of a set system ([ n ] , R ) with f R ( b ) = ζ ( b ) − 1 is E x ( n, b, ζ ( b ) − b − 2) + n + 1 . Pr o of. By Lemma 1 it sues to pro v e the statemen t for ideal set systems. Let ([ n ] , R ) b e an ideal set system with f R ( k ) < ζ ( b ) − 1 and maximal size. If R on tains some range A of size 3 , then | R | A | = 8 . No w, write b = 3 + 2 j + s with s ∈ { 0 , 1 } . Let B denote the set A augmen ted b y j pairs of elemen ts that b elong to R , and one single elemen t of R if s = 1 . The set B has size b and the trae of R on B has size at least 8 + 3 j + s = ζ ( b ) . Th us, if f R ( b ) < ζ ( b ) w e get that R on tains no triple, and an th us b e deomp osed in to R = { ∅ } ∪ V ∪ E , where V are the singletons and E the pairs in R ; all the former the verti es of R and the latter its e dges . If some elemen t x ∈ [ n ] is not a singleton of V then it is on tained in no range of R , and w e an delete it without hanging the size of R ; this on tradits the maximalit y of R . No w, notie that the trae of R on an y b elemen ts on tains at most f R ( b ) − b − 1 = ζ ( b ) − b − 2 edges, sine it on tains the empt y set and ea h of the b v erties. Con v ersely , let G = ([ n ] , E ) b e a graph whose indued graph on an y b v erties has at most ζ ( b ) − b − 2 edges. If R = {∅ } ∪ [ n ] ∪ E then the set system ([ n ] , R ) satises f R ( b ) < ζ ( b ) and the statemen t follo ws. A graph whose indued subgraphs on an y m v erties ha v e at most k < ⌊ m 2 4 ⌋ edges annot on tain a K ⌊ k 2 ⌋ , ⌊ k 2 ⌋ , and th us, b y the K ® v ári-Sós-T urán Theorem, has at most E x ( n, m, k ) = O n 2 − 1 ⌊ k 2 ⌋ edges. It follo ws that: E x ( n, 4 , 3) = E x ( n, 5 , 5) = O ( n √ n ) . The lassial onstrutions yielding bipartite graphs on n v erties with Θ( n √ n ) edges and no K 2 , 2 sho w that this b ound is b est p ossible. Sine ζ (4) = 9 , w e get that the family of gro wth rates obtained b y the onditions f R ( b ) = k do es not only on tain p olynomial gro wth with in teger exp onen ts: Corollary 4. The lar gest set system ([ n ] , R ) with f R (4) = 8 or f R (5) = 10 has size | R | = Θ ( n √ n ) . Note that Lemma 3 an b e extended in to an equiv alene of Question 1 and Dira's problem on r -regular h yp ergraphs for arbitrary large r . RR n ° 7154 8 Che ong, Go ao & Ni aud 3 F amilies of p erm utations In this setion w e giv e b ounds on the size of a family F of p erm utations on [ n ] with a giv en φ F ( b ) . Redution to set systems. An inversion of a p erm utation σ on [ n ] is a pair of elemen ts i < j su h that σ − 1 ( i ) > σ − 1 ( j ) . The distinguishing p air of t w o p erm utations σ 1 and σ 2 is the lexiographially smallest pair ( i, j ) ⊂ [ n ] that app ears in dieren t orders in σ 1 and σ 2 , i.e. is an in v ersion for one but not for the other. If F is a family of p erm utations on [ n ] w e let I F denote the set of distinguishing pairs of pairs of p erm utations from F . Giv en a p erm utation σ ∈ F , w e let R ( σ ) denote the set of elemen ts of I F that are in v ersions of σ , and let R ( F ) = { R ( σ ) | σ ∈ F } . Observ e that ( I F , R ( F )) is a range spae and that R is a one-to-one map b et w een F and R ( F ) . In partiular | F | = | R ( F ) | . Lemma 5. f R ( F ) ( ⌊ m 2 ⌋ ) ≤ φ F ( m ) and | I F | ≤ E x ( n, m, φ F ( m ) − 1) . Pr o of. Consider b = ⌊ m 2 ⌋ elemen ts ( p 1 , . . . , p b ) in I F and assume there exists k ranges R ( σ 1 ) , . . . , R ( σ k ) with distint traes on { p 1 , . . . , p b } . Then the re- stritions of σ 1 , . . . , σ k on X = ∪ 1 ≤ i ≤ b p i m ust also b e pairwise distint. Th us, φ F ( m ) ≥ k whenev er f R ( F ) ( ⌊ m 2 ⌋ ) ≥ k , and the statemen t follo ws. Let s ( t ) denote the maxim um n um b er of distinguishing pairs in a family of t p erm utations (on [ n ] ). F rom s (2) = 1 and s ( t ) ≤ 1 + max 1 ≤ i ≤ t − 1 { s ( i ) + s ( t − i ) } , w e get that s ( t ) ≤ t − 1 b y a simple indution. This implies that in the graph G = ([ n ] , I F ) , an y m v erties span at most φ F ( m ) − 1 edges, and it follo ws that | I F | ≤ E x ( n, m, φ F ( m ) − 1) , whi h onludes the pro of. A sub quadrati I F is not alw a ys p ossible: ev ery pair is a distinguishing pair of the family of all p erm utations on [ n ] that restrit to the iden tit y on some ( n − 1) -tuple. F or that family , φ F ( m ) = ( m − 1) 2 + 1 . Main transitions. W e an no w outline the main transitions in the gro wth rate of families of p erm utations aording to the v alue of φ F ( m ) . Let b = ⌊ m 2 ⌋ . If φ F ( m ) ≤ ⌊ m 2 ⌋ then, b y Lemma 5 , f R ( F ) ( b ) ≤ b and Theorem 2 with i = 0 yields that | F | = | R ( F ) | = O (1) . Assume that ⌊ m 2 ⌋ < φ F ( m ) < 2 ⌊ m 2 ⌋ . Then, b y Lemma 5 , f R ( F ) ( b ) < 2 b and Theorem 2 with i = 1 yields that | F | = | R ( F ) | = O ( | I F | ) = O ( E x ( n, m, m − 2)) = O ( n ) . A mat hing lo w er b ound is giv en b y the family F 1 : all p erm utations on [ n ] that dier from the iden tit y b y the transp osition of a single pair of the form (2 i, 2 i + 1) , of size 1 + ⌊ n 2 ⌋ and with φ F 1 ( m ) = ⌊ m 2 ⌋ + 1 . INRIA Set Systems and F amilies of Permutations with Smal l T r a es 9 If φ F ( m ) < 2 ⌊ m 2 ⌋ then, b y Lemma 5, f R ( F ) ( b ) < 2 b and ( I F , R ( F )) has V C-dimension at most b − 1 . It follo ws, from Sauer's Lemma, that | F | = | R ( F ) | = O ( | I F | b − 1 ) , and sine | I F | = O ( n 2 ) , w e get that | F | is O n 2 ⌊ m 2 ⌋− 2 . If φ F ( m ) ≥ 2 ⌊ m 2 ⌋ then the family F 2 : all p erm utations on [ n ] that dier from the iden tit y b y the transp osition of an y n um b er of pairs of the form { 2 i, 2 i + 1 } , of size 2 ⌊ n 2 ⌋ and with φ F 2 ( m ) = 2 ⌊ m 2 ⌋ sho ws that | F | an b e exp onen tial in n . If φ F ( m ) = m then | I F | = O ( E x ( n, m, m − 1)) , whi h is sup erlinear and O ( n 1+ 1 ⌊ m 2 ⌋ ) [11 ℄. W e ha v e not found an y example sho wing that F ould ha v e sup erlinear size. The main transitions are summarized in T able 2. φ F ( m ) ≤ ⌊ m 2 ⌋ ⌊ m 2 ⌋ < φ F ( m ) < 2 ⌊ m 2 ⌋ 2 ⌊ m 2 ⌋ ≤ φ F ( m ) < 2 ⌊ m 2 ⌋ 2 ⌊ m 2 ⌋ ≤ φ F ( m ) | F | Θ(1) Θ( n ) Ω( n ) and O n 2 ⌊ m 2 − 2 ⌋ Ω(2 ⌊ n 2 ⌋ ) T able 2: Maxim um size of a family F of p erm utations as a funtion of φ F ( m ) . Exp onen tial upp er b ounds. Raz [14 ℄ pro v ed that if φ F (3) ≤ 5 then | F | has size at most exp onen tial in n . The follo wing simple observ ation deriv es a similar b ounds for a few other v alues of φ F ( m ) . Lemma 6. If | F | is at most exp onential whenever φ F ( m − 1) ≤ k − 1 then | F | is at most exp onential whenever φ F ( m ) ≤ k . Pr o of. Let T ( n, m, k ) denote the maxim um size of a family F su h that φ F ( m ) ≤ k . Assume that φ F ( m ) = k = φ F ( m − 1) as otherwise the statemen t trivially holds. Let X ∈ [ n ] m − 1 su h that F | X = { σ 1 , . . . , σ k } has size k , and let: F i = { σ ∈ F | σ | X = σ i } . Observ e that F is the disjoin t union of the F i . Sine φ F ( m ) = k , for an y e ∈ [ n ] \ X and for an y i = 1 , . . . , k , there exists a unique p erm utation in ( F i ) | X ∪{ e } that restrits to σ i on X . In other w ords, for ev ery elemen t in [ n ] \ X , the set X ∪ { e } app ears in the same order in all p erm utations of F i . It follo ws that | F i | = | ( F i ) | [ n ] \ X | , that is, deleting X do es not derease the size of ea h F i onsidered individually although it ma y derease the size of F . No w, let G i = ( F i ) | [ n ] \ X and onsider the set system ([ n ] \ X , G i ) . If φ G i ( m − 1) ≤ k − 1 then | G i | ≤ T ( n − m +1 , m − 1 , k − 1) , and otherwise φ G i ( m − 1) = k and w e reurse. Altogether, w e ha v e the reursion T ( n, m, k ) ≤ k max ( T ( n − m + 1 , m − 1 , k − 1) , T ( n − m + 1 , m, k )) , and it follo ws that if T ( n, m − 1 , k − 1) is at most exp onen tial, so is T ( n, m, k ) . It then follo ws, with Raz's result, that | F | is at most exp onen tial whenev er φ F ( m ) ≤ m + 2 . T able 3 tabulates our results for small v alues of m and φ F ( m ) , using the urren tly b est kno wn b ounds on E x ( n, k , µ ) w e are a w are of [11 ℄. RR n ° 7154 10 Che ong, Go ao & Ni aud k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 k = 10 m = 2 n ! - - - - - - - - m = 3 2 Θ( n ) 2 Θ( n ) 2 Θ( n ) 2 Θ( n ) n ! - - - - m = 4 2 ⌊ 2 n 3 ⌋ + 1 2 Θ( n ) 2 Θ( n ) 2 Θ( n ) 2 Ω( n ) 2 Ω( n ) 2 Ω( n ) 2 Ω( n ) m = 5 2 ⌊ n 2 ⌋ + 1 2 Θ( n ) 2 Θ( n ) 2 Θ( n ) 2 Θ( n ) 2 Ω( n ) 2 Ω( n ) 2 Ω( n ) m = 6 2 3 Θ( n ) Θ( n ) O ( n 3 ) O ( n 3 ) 2 Θ( n ) 2 Ω( n ) 2 Ω( n ) m = 7 2 3 Θ( n ) Θ( n ) O ( n 2 ) O ( n 3 ) 2 Θ( n ) 2 Θ( n ) 2 Ω( n ) m = 8 2 3 4 Θ( n ) Θ( n ) Θ( n ) O ( n 21 / 8 ) O ( n 7 / 2 ) O ( n 7 / 2 ) m = 9 2 3 4 Θ( n ) Θ( n ) Θ( n ) O ( n 2 ) O ( n 7 / 2 ) O ( n 7 / 2 ) m = 10 2 3 4 5 Θ( n ) Θ( n ) Θ( n ) Θ( n ) O ( n 27 / 10 ) T able 3: Maxim um size of a family F of p erm utations on [ n ] with φ F ( m ) = k . 4 Conlusion A natural op en question is the tigh tening of the b ounds for b oth Questions 1 and 2 . In partiular, the rst ase where Lemma 5 no longer guaran tees that the redution from p erm utations to set systems leads to a ground set with linear size is φ F ( m ) = m ; do es that ondition still imply that | F | is O ( n ) when m is large enough? Raz's generalization of the Stanley-Wilf onjeture, that is, whether φ F ( m ) < m ! implies that | F | is exp onen tial in n , also app ears to b e a hallenging question. Can it b e ta kled b y a normalization te hnique similar to Lemma 1? A line in terseting a olletion C of pairwise disjoin t on v ex sets in R d in- dues t w o p erm utations, one rev erse of the other, orresp onding to the order in whi h ea h orien tation of the line meets the set. The pair of these p erm uta- tions is alled a ge ometri p ermutation of C . One of the main op en questions in geometri transv ersal theory [19 ℄ is to b ound the maxim um n um b er of ge- ometri p erm utations of a olletion of n pairwise disjoin t sets in R d (see for instane [2 , 3 , 8, 16 ℄). W e an pi k from ea h geometri p erm utation one of its el- emen ts so that the resulting family F has the follo wing prop ert y: if an y m mem- b ers of C ha v e at most k distint geometri p erm utations then φ F ( m − 2) ≤ k . One in teresting question is whether b ounds su h as the one w e obtained ould lead to new results on the geometri p erm utation problem. A kno wledgmen ts The authors thank Olivier Devillers and Csaba Tóth for their helpful ommen ts. Referenes [1℄ N. Alon. Densit y of sets of v etors. Disr ete Mathematis , 46:199202, 1983. [2℄ B. Arono v and S. Smoro dinsky . On geometri p erm utations indued b y lines transv ersal through a xed p oin t. Disr ete & Computational Ge ome- try , 34:285294, 2005. INRIA Set Systems and F amilies of Permutations with Smal l T r a es 11 [3℄ A. Asino wski and M. Kat halski. The maximal n um b er of geometri p er- m utations for n disjoin t translates of a on v ex set in R 3 is ω ( n ) . Disr ete & Computational Ge ometry , 35:473480, 2006. [4℄ B. Bollobás. Combinatoris . Cam bridge Univ ersit y Press, 1986. [5℄ B. Bollobás and A. J. Radlie. Defet Sauer results. Journal of Combi- natorial The ory. Series A , 72(2):189208, 1995. [6℄ H. Bronnimann and M.T. Go o dri h. Almost optimal set o v ers in nite V C-dimension. Disr ete & Computational Ge ometry , 14:463479, 1995. [7℄ B. Chazelle. The disr ep any metho d: r andomness and omplexity . Cam- bridge Univ ersit y Press, 1986. [8℄ O. Cheong, X. Goao , and H.-S. Na. Geometri p erm utations of disjoin t unit spheres. Computational Ge ometry: The ory & Appli ations , 30:253 270, 2005. [9℄ P . F rankl. On the trae of nite sets. Journal of Combinatorial The ory. Series A , 34:4145, 1983. [10℄ Z. Füredi and J. P a h. T raes of nite sets: Extremal problems and geo- metri appliations. In Extr emal Pr oblems for Finite Sets , v olume 3, pages 255282. Boly ai So iet y , 1994. [11℄ J. R. Griggs and M. Simono vits , G. Rubin Thomas. Extremal graphs with b ounded densities of small subgraphs. Journal of Gr aph The ory , 3:185207, 1998. [12℄ Donald E. Kn uth. A rt of Computer Pr o gr amming, V olume 1: F undamental A lgorithms (3r d Edition) . A ddison-W esley Professional, 1997. [13℄ A. Marus and G. T ardos. Exluded p erm utation matries and the Stanley Wilf onjeture. Journal of Combinatorial The ory. Series A , 107(1):153 160, 2004. [14℄ R. Raz. VC-dimension of sets of p erm utations. Combinatori a , 20:255, 2000. [15℄ N. Sauer. On the densit y of families of sets. Journal of Combinatorial The ory. Series A , 13:145147, 1972. [16℄ M. Sharir and S. Smoro dinsky . On neigh b ors in geometri p erm utations. Disr ete Mathematis , 268:327335, 2003. [17℄ S. Shelah. A om binatorial problem; stabilit y and order for mo dels and theories in innitary languages. Pai J. Math , 41(1):247261, 1972. [18℄ V. N. V apnik and A. Y a. Cherv onenkis. On the uniform on v ergene of relativ e frequenies of ev en ts to their probabilities. The ory Pr ob ab. Appl. , 16(2):264280, 1971. [19℄ R. W enger. Helly-t yp e theorems and geometri transv ersals. In Jaob E. Go o dman and Joseph O'Rourk e, editors, Handb o ok of Disr ete & Compu- tational Ge ometry , hapter 4, pages 7396. CR C Press LLC, Bo a Raton, FL, 2nd edition, 2004. RR n ° 7154 Centre de recherche INRIA Nancy – Grand Est LORIA, T echnop ôle de Nancy-Brabois - C ampu s scientifique 615, rue du Jardin Botanique - BP 101 - 5460 2 V illers-lès-Nancy Cedex (France) Centre de recherc he INRIA Bordeaux – Sud Ouest : Domaine Univ ersitaire - 351, cours de la Libération - 33405 T alenc e Cedex Centre de recherc he INRIA Grenoble – Rhône-Alpes : 655, ave nue de l’Europe - 38334 Montbonno t Saint-Ismier Centre de recherc he INRIA Lille – Nord Europe : Parc Scienti fique de la Haute Borne - 40, aven ue Halley - 59650 V illeneuv e d’Ascq Centre de recherc he INRIA Paris – Rocquenc ourt : Domaine de V oluce au - Rocquencou rt - BP 105 - 78153 Le Chesnay Cedex Centre de recherc he INRIA Rennes – Bretagne Atlantique : IRISA, Campus univ ersitai re de Beaulieu - 35042 Rennes Cedex Centre de recherc he INRIA Saclay – Île-de-Franc e : Parc Orsay Univ ersité - ZA C des V ignes : 4, rue Jacques Monod - 91893 Orsay Cedex Centre de recherc he INRIA Sophia Antipolis – Méditerra née : 2004, route des Luciole s - BP 93 - 06902 Sophia Antipolis Cedex Éditeur INRIA - Domaine de V olucea u - Rocquenc ourt, BP 105 - 78153 Le Chesnay Cedex (France) http://www.inria.fr ISSN 0249 -6399
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