Exemal poblems fo -paie and -coloable hypegaphs Dhuv Mubayi John Talbo June, 007 Absac Fix ineges and an -unifom hypegaph F. We pove ha he maximum numbe of edges in a -paie -unifom hypegaph on n veices ha conains no copy ( of F is c n,f ) + o(n ), whee c,f can be deemined by a finie compuaion. We explicily define a sequence F 1, F,... of -unifom hypegaphs, and pove ha he maximum numbe of edges in a -chomaic -unifom hypegaph on n veices ( conaining no copy of F i is α n,,i ) + o(n ), whee α,,i can be deemined by a finie compuaion fo each i 1. In seveal cases, α,,i is iaional. The main ool used in he poofs is he Lagangian of a hypegaph. 1 Inoducion An -unifom hypegaph o -gaph is a pai G = (V, E) of veices, V, and edges E ( V ), in paicula a -gaph is a gaph. We denoe an edge {v 1, v,..., v } by v 1 v v. Given -gaphs F and G we say ha G is F -fee if G does no conain a copy of F. The maximum numbe of edges in an F -fee -gaph of ode n is ex(n, F ). Fo = and F = K s (s 3) Depamen of Mahemaics, Saisics and Compue Science, Univesiy of Illinois, Chicago, IL 60607, and Depamen of Mahemaical Sciences, Canegie-Mellon Univesiy, Pisbugh, PA 1513. Email: mubayi@mah.uic.edu. Reseach suppoed in pa by NSF gans DMS-040081 and 0653946 and an Alfed P. Sloan Reseach Fellowship. Depamen of Mahemaics, UCL, London, WC1E 6BT, UK. Email: albo@mah.ucl.ac.uk. This auho is a Royal Sociey Univesiy Reseach Fellow. 1
his numbe was deemined by Tuán [T41] (ealie Manel [M07] found ex(n, K 3 )). Howeve in geneal (even fo = ) he poblem of deemining he exac value of ex(n, F ) is beyond cuen mehods. The coesponding asympoic poblem is o deemine he Tuán densiy ex(n,f ) of F, defined by π(f ) = lim n (his always exiss by a simple aveaging agumen ( n ) due Kaona e al. [KNS64]). Fo -gaphs he Tuán densiy is deemined by he chomaic numbe of he fobidden subgaph F. The explici elaionship is given by he following fundamenal esul. Theoem 1 (Edős Sone Simonovis [ES46],[ES66]). If F is a -gaph hen π(f ) = 1 1. χ(f ) 1 When 3, deemining he Tuán densiy is difficul, and hee ae only a few exac esuls. Hee we conside some closely elaed hypegaph exemal poblems. Call a hypegaph H -paie if is veex se can be paiioned ino classes, such ha evey edge has a mos one veex in each class. Call H -coloable, if is veex se can be paiioned ino classes so ha no edge is eniely conained wihin a class. Definiion. Fix, and an -gaph F. Le ex (n, F ) (ex (n, F )) denoe he maximum numbe of edges in a -paie (-coloable) -gaph on n veices ha conains no copy of F. The -paie Tuán densiy of F is π (F ) = lim n ex (n, F )/ ( n ) and he -chomaic Tuán densiy of F is π (F ) = lim n ex (n, F )/ ( n ). Noe ha i is easy o show ha hese limis exis. In his pape, we deemine π (F ) fo all -gaphs F and deemine π (F ) fo an infinie family of -gaphs (peviously no nonivial value of π (F ) was known). In many cases ou examples yield iaional values of π (F ). Fo he usual Tuán densiy, π(f ) has no been poved o be iaional fo any F, alhough hee ae seveal conjecues saing iaional values. In ode o descibe ou esuls, we need he concep of G-colouings which we inoduce now. If F and G ae hypegaphs (no necessaily unifom) hen F is G-colouable if hee exiss c : V (F ) V (G) such ha c(e) E(G) wheneve e E(F ). In ohe wods, F is G-colouable if hee is a homomophism fom F o G. Le K () K () denoe he complee -gaph of ode. Then an -gaph F is -paie if F is -colouable, and F is -colouable if i is H () -colouable whee H () is he (in geneal non-unifom) hypegaph consising of all subses A {1,,..., } saisfying A ). The chomaic numbe of F is χ(f ) = min{ 1 : F is -colouable}. Noe ha while a
-gaph is -colouable iff i is -paie his is no longe ue fo 3, fo example K (3) 4 is -colouable bu no -paie o 3-paie. Le G () denoe he collecion of all -veex -gaphs. A ool which has poved vey useful in exemal gaph heoy and which we will use lae is he Lagangian of an -gaph. Le S = { x R : x i = 1, x i 0 fo 1 i }. i=1 If G G () and x S hen we define λ(g, x) = x v1 x v x v. v 1 v v E(G) The Lagangian of G is max x S λ(g, x). The fis applicaion of he Lagangian o exemal gaph heoy was due o Mozkin and Sauss who gave a new poof of Tuán s heoem. We ae now eady o sae ou main esul. Theoem 3. If F is an -gaph and hen π (F ) = max{!λ(g) : G G () and F is no G-colouable}. As an example of Theoem 3, suppose ha = 4, = 3, and F = K (3) 4. Le H denoe he unique 3-gaph wih fou veices and hee edges. Now F is F -coloable, bu i is no H-coloable, and he lagangian λ(h) of H is 4/81, achieved by assigning he degee hee veex a weigh of 1/3 and he ohe hee veices a weigh of /9. Consequenly, Theoem 3 says ha he maximum numbe of edges in an n-veex 4-paie 3-gaph conaining no copy of K (3) 4 is (8/7) ( n 3) + o(n 3 ). This is clealy achievable, by he 4-paie 3-gaph wih pa sizes n/3, n/9, n/9, n/9, wih all possible iples beween hee pas ha include he lages (of size n/3), and no iples beween he hee small pas. Chomaic Tuán densiies wee peviously consideed in [T07] whee hey wee used o give an impoved uppe bound on π(h), whee H is defined in he pevious paagaph. Howeve no non-ivial chomaic Tuán densiies have peviously been deemined. each we ae able o give an infinie sequence of -gaphs whose -chomaic Tuán densiies ae deemined exacly. Fo l and define β,,l := max{λ(g) : G is a -coloable -gaph on l veices}. 3 Fo
I seems obvious ha β,,l is achieved by he -chomaic -gaph of ode l wih all colo classes of size l/ o l/ and all edges pesen excep hose wihin he classes. Noe ha if l hen his would give β,,l = (( ) l ( l/ )) 1 l. Howeve, we ae only able o pove his fo =, 3. If he above saemen is ue, hen β,,l can be compued by calculaing he maximum of an explici polynomial in one vaiable ove he uni ineval. In any case i can be obained by a finie compuaion (fo fixed,, l). Le α,,l =!β,,l. Theoem 4. Fix l. Le L () be he -gaph obained fom he complee gaph K by enlaging each edge wih a se of new veices. If hen whee α,,l is defined above. π (L () ) = α,,l The emainde of he pape is aanged as follows. In he nex secion we pove Theoem 3 and in he las secion we pove Theoem 4 and he saemens abou compuing β,,l, fo =, 3. Poof of Theoem 3 If G G () and x = (x 1,..., x ) Z + hen he x-blow-up of G is he -gaph G( x) consuced fom G by eplacing each veex v by a class of veices of size x v and aking all edges beween any classes coesponding o an edge of G. Moe pecisely we have V (G( x)) = X 1 X, X i = x i and E(G( x)) = {{v i1 v i v i } : v ij X ij, {i 1 i i } E(G)}. If x = (s, s,..., s) and G = K () hen G( x) is he complee -paie -gaph wih class size s, denoed by K () (s). Noe ha if F and G ae boh -gaphs hen F is G-colouable iff hee exiss x Z + such ha F G( x). An -gaph G is said o be coveing if each pai of veices in V (G) is conained in a common edge. If W V and G is an -gaph wih veex V hen G[W ] is he induced subgaph of G fomed by deleing all veices no in W and emoving all edges conaining hese veices. 4
Lemma 5 (Fankl and Rödl [FR84]). If G is an -gaph of ode n hen hee exiss y S n wih λ(g) = λ(g, y), such ha if P = {v V (G) : y v > 0} hen G[P ] is coveing. Supesauaion fo odinay Tuán densiies was shown by Edős [E71]. The poof fo G- chomaic Tuán densiies is essenially idenical bu fo compleeness we give i. We equie he following classical esul. Theoem 6 (Edős [E64]). If and 1 hen ex(n, K () ()) = O(n λ, ), wih λ, > 0. Lemma 7 (Supesauaion). Fix. If G is an -gaph, H is a finie family of - gaphs, s 1 and s = (s, s,..., s) hen π (H( s)) = π (H) (whee H( s) = {H( s) : H H}). Poof: Le p = max{ V (H) : H H}. By adding isolaed veices if necessay we may suppose ha evey H H has exacly p veices. Fis we claim ha if F is an n-veex -gaph wih densiy a leas α + ɛ, whee α, ɛ > 0, and m n hen a leas ɛ ( n m) of he m-veex induced subgaphs of F have densiy a leas α + ɛ. To see his noe ha if i fails o hold hen ) ( ) n ( )( ) ( ) ( n m n m ( n m (α + ɛ) which is impossible. W ( V (F ) m ) e(f [W ]) < ɛ m + (1 ɛ) m (α + ɛ) Le ɛ > 0 and suppose ha F is an n-veex -gaph wih densiy a leas π (H) + ɛ. We need o show ha if n sufficienly lage hen F conains a copy of H( s). Le m m(ɛ) be sufficienly lage ha any -paie m-veex -gaph wih densiy a leas π (H)+ɛ conains a copy of some H H. We say ha W ( ) V (F ) m is good if F [W ] conains a copy of some H H. By he claim a leas ɛ ( ( n m) m-ses ae good, so if δ = ɛ/ H hen a leas δ n m) m-ses conain a fixed H H. Thus he numbe of p-ses U V (F ) such ha F [U] H is a leas δ ( ) n m ) = δ( ) n p ( m ). (1) p ( n p m p Le J be he p-gaph wih veex se V (F ) and edge se consising of hose p-ses U V (F ) such ha F [U] H. Now, by Theoem 6, ex (n, K () ()) ex(n, K () ()) = O(n λ, ), whee λ, > 0. Hence (1) implies ha fo any if n is sufficienly lage hen K p (p) () J. ), 5
Finally conside a colouing of he edges of K p (p) () wih p! diffeen colous, whee he colou of he edge is given by he ode in which he veices of H ae embedded in i. By Ramsey s heoem if is sufficienly lage hen hee is a copy of K p (p) (s) wih all edges he same colou. This yields a copy of H ( s) in F as equied. Poof of Theoem 3. Le α, = max{!λ(g) : G G () is well-defined since G () ( ) is finie.) and F is no G-colouable}. (This If G G () and F is no G-colouable hen fo any x Z + we have F G( x). Le y S saisfy λ(g, y) = λ(g). Fo n 1 le x n = ( y 1 n,..., y n ) Z +. If G n = G( x n ) hen e(g n ) lim ( n n =!λ(g). ) Moeove since each G n is F -fee, -paie and of ode a mos n we have π (F )!λ(g). Hence π (F ) α,. Le H(F ) = {H G () : F is H-colouable}. I is sufficien o show ha π (H(F )) α,. () Indeed, if we assume ha () holds, hen le s 1 be minimal such ha evey H H(F ) saisfies F H( s), whee s = (s, s,..., s). (Noe ha s exiss since F is H-colouable fo evey H H(F )). Now by supesauaion (Lemma 7) if ɛ > 0, hen any -paie -gaph G n wih n n 0 (s, ɛ) veices and densiy a leas α, + ɛ will conain a copy of H( s) fo some H H(F ). In paicula G n conains F and so π (F ) α,. Le π (H(F )) = γ and ɛ > 0. If n is sufficienly lage hee exiss an H(F )-fee, -paie -gaph G n of ode n saisfying!e(g n ) γ ɛ. n Taking y = (1/n, 1/n,..., 1/n) S n we have!λ(g n )!λ(g n, y) =!e(g n) n γ ɛ. Now Lemma 5 implies ha hee exiss z S n saisfying λ(g n ) = λ(g n, z) and G n [P ] is coveing whee P = {v V (G) : z v > 0}. 6
Since G n is -paie, we conclude ha G n [P ] has a mos veices. Moeove, G n is H(F )- fee and so G n [P ] H(F ). Thus F is no G n [P ]-coloable, and we have γ ɛ!λ(g n [P ]) α,. Thus π (H(F )) α, + ɛ fo all ɛ > 0. Hence () holds and he poof is complee. 3 Infiniely many chomaic Tuán densiies Fo l, le K () l be he family of -gaphs wih a mos ( l ) edges ha conain a se S, called he coe, of l veices, wih each pai of veices fom S conained in an edge. Noe ha L () K(). We need he following Lemma ha was poved in [M06]. Fo compleeness, we epea he poof below. Lemma 8. If K K (), s = ( ) () + 1 and s = (s, s,..., s) hen L K( s). Poof. We fis show ha L () L(( ) () + 1) fo evey L K. Pick L K(), and le L = L( ( ) + 1). Fo each veex v V (L), suppose ha he clones of v ae v = v 1, v,..., v ( )+1. In paicula, idenify he fis clone of v wih v. Le S = {w 1,..., w } V (L) be he coe of L. Fo evey 1 i < j l + 1, le E ij L wih E ij {w i, w j }. Replace each veex z of E ij {w i, w j } by z q whee q > 1, o obain an edge E ij L. Coninue his pocedue fo evey i, j, making sue ha wheneve we encoune a new edge i inesecs he peviously encouneed edges only in L. Since he numbe of clones is ( ) + 1, his pocedue can be caied ou successfully and esuls in a copy of L () wih coe S. Theefoe L() L = L( ( ) + 1). Consequenly, Lemma 7 implies ha π(l () ) π(k() ). Poof of Theoem 4. Le l and. We will pove ha π (K () ) = α,,l. (3) The heoem will hen follow immediaely fom Lemmas 7 and 8. Le B,,l = {G : G is a -colouable K () -fee -gaph}. Claim. max{λ(g) : G B,,l } = β,,l = α,,l /!. Poof of Claim. If G B,,l has ode n hen Lemma 5 implies ha hee is y S n such ha λ(g) = λ(g, y) wih G[P ] coveing, whee P = {v V (G) : y v > 0}. Since G is 7
K () -fee, we conclude ha P = p l. Hence hee is H B,,l such ha λ(h) = λ(g) and H has ode a mos l. Consequenly, max{λ(g) : G B,,l } β,,l. Fo he ohe inequaliy, we jus obseve ha an l veex -gaph mus be K () -fee. Now we can quickly complee he poof of he heoem by poving (3). Fo he uppe bound, obseve ha if G B,,l has ode n hen by he Claim e(g) n λ(g) α,,l! and so π (K () ) α,,l. Fo he lowe bound, suppose ha G B,,l has ode p and saisfies λ(g) = β,,l. Then hee exiss y S p such ha λ(g, y) = λ(g) = β,,l. Fo n p define y n = ( y 1 n,..., y p n ). Now {G( y n )} n=p is a sequence of -colouable K () -fee -gaphs and hence π (K () ) lim e(g n ) ( n n =!λ(g) = α,,l. ) Now we pove ha β,,l can be compued by only consideing maximum -coloable -gaphs wih almos equal pa sizes when =, 3. The case = follows ivially fom Lemma 5 so we conside he case = 3. Theoem 9. Fix l. Then β 3,,l is achieved by he -chomaic 3-gaph of ode l wih all colo classes of size l/ o l/ and all edges pesen excep hose wihin he classes. Remak: Noe ha if l hen his implies ha β 3,,l = ( ( ) ( l 3 l/ ) 3 ) 1 Poof. Le G be a -chomaic 3-gaph of ode l. We may suppose (by adding edges as equied) ha V (G) = V 1 V V and ha all edges no conained in any V i ae pesen. We may also suppose ha V 1 V V. Le x S p saisfy λ(g, x) = λ(g). If v, w V i and x v > x w hen fo a suiable choice of δ > 0 we can incease λ(g, x) by inceasing x w by δ and deceasing x v by δ. Hence we may suppose ha hee ae x 1,..., x such ha all veices in V i eceive weigh x i. Le l = b + c, 0 c <. To complee he poof we need o show ha all of he V i have ode b o b + 1. Suppose, fo a conadicion, ha hee exis V i and V j wih a i = V i, a j = V j and a i a j +. Moving a veex v fom V i o V j and inseing all new allowable edges (i.e. hose which conain v and veices fom V i \{v}) while deleing any edges which l 3. 8
now lie in V j we canno incease λ(g, x). This implies ha ( ) ( ) aj x i x ai 1 j x 3 i, (4) and so in paicula x i < x j. Le G denoe his new -colouable 3-gaph. We give a new weighing y fo G by seing a i x i /(a i 1), v V i, y v = a j x j /(a j + 1), v V j, x k, v V k and k i, j. I is easy o check ha y S l is a legal weighing fo G. We will deive a conadicion by showing ha λ( G) λ( G, y) > λ(g, x) = λ(g). If w = a i x i + a j x j = (a i 1)y i + (a j + 1)y j hen (( ) ( ) λ( G, ai 1 y) λ(g, x) = (1 w) yi aj + 1 + yj + (a i 1)(a j + 1)y i y j ( ) ( ) ( ) ai x aj i )x j ai 1 a i a j x i x j + (a j + 1)yi y j + ( ) ( ) ( ) aj + 1 (a i 1)y i yj ai a j x aj i x j a i x i x j = (1 w) ( aj x j a j + 1 a ix i a i 1 Using (4) i is easy o check ha his is sicly posiive. ) + a ia j x i x j ( xj a j + 1 Coollay 10. The -chomaic Tuán densiy can ake iaional values. x ) i. a i 1 Poof. We conside β 3,,k fo k 3. In fac, we focus of β 3,,6, he maximum densiy of a -chomaic 3-gaph ha conains no copy of K (3) 6. By he pevious Theoem, his is 6 imes he lagangian of he 3-gaph wih veex se {a, a, a, b, b } and all edges pesen excep {a, a, a }. Assigning weigh x o he a s and weigh y o he b s, we mus maximize 6(6x y + 3xy ) subjec o 3x + y = 1 and 0 x 1/3. A sho calculaion shows ha he choice of x ha maximizes his expession is ( 13 )/9, and his esuls in an iaional value fo he lagangian. Simila compuaions hold fo lage k as well. 9
Refeences [E64] P. Edős, On exemal poblems of gaphs and genealized gaphs, Isael J. Mah. (1964), 183 190. [E71] P. Edős, On some exemal poblems on -gaphs, Disc. Mah. 1 (1971), 1 6. [ES66] [ES46] P. Edős and M. Simonovis, A limi heoem in gaph heoy, Sudia Sci. Ma. Hung. Acad. 1 (1966), 51 57. P. Edős and A.H. Sone, On he sucue of linea gaphs, Bull. Ame. Mah. Soc. 5(1946), 1087 1091. [FR84] P. Fankl and V. Rödl, Hypegaphs do no jump, Combinaoica 4 (1984), 149 159. [KNS64] G. Kaona, T. Nemez and M. Simonovis, On a poblem of Tuan in he heoy of gaphs (in Hungaian) Ma. Lapok 15 (1964), 8 38. [M07] W. Manel, Poblem 8, Wiskundige Opgaven, 10 (1907), 60-61. [MS65] [M06] [T07] T. Mozkin and E. Sauss. Maxima fo gaphs and a new poof of a heoem of Tuan, Canadian Jounal of Mahemaics, 17:533 540, 1965. D. Mubayi, A hypegaph exension of Tuán s heoem, J. Combin. Theoy, Se. B 96 (006) 1 134. J. Talbo, Chomaic Tuán poblems and a new uppe bound fo he Tuán densiy of K4. Euop. J. Comb. (o appea) (007?) [T41] P. Tuán, On an exemal poblem in gaph heoy, Ma. Fiz. Lapok 48 (1941) 10