Discrete Mathematics 309 (009) 86 869 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/isc Profile vectors in the lattice of subspaces Dániel Gerbner a, Balázs Patós b, a Department of Information Systems, Eötvös University, Pázmány Péter sétány /B, Buapest, 7, Hungary b Department of Mathematics an its Applications, Central European University, Náor u 9, Buapest, 05, Hungary a r t i c l e i n f o a b s t r a c t Article history: Receive 9 September 006 Receive in revise form 7 July 008 Accepte 0 July 008 Available online 5 August 008 Keywors: Profile vectors Lattice of subspaces Intersecting families The profile vector f (U) R n of a family U of subspaces of an n-imensional vector space over GF(q) is a vector of which the ith coorinate is the number of subspaces of imension i in the family U(i = 0,,, n) In this paper, we etermine the profile polytope of intersecting families (the convex hull of the profile vectors of all intersecting families of subspaces) 008 Elsevier B All rights reserve Introuction Many problems in extremal set theory consier a set A of families of subsets of an n-element set all having some fixe property All subsets F possess a weight w(f) epening only on F, the size of F an we as for the family F with the largest weight w(f ) = F F w(f) (Note, that asing for the family with largest size is equivalent to asing for the family with largest weight for the constant weight ) The basic tool for ealing with all ins of weight functions simultaneously is the profile vector f (F ) of a family F which is efine by f (F ) i = {F F : F = i} (i = 0,,, n) With this notation the weight of a family for a given weight function w is simply the inner prouct of the weight vector an the profile vector Therefore, as we now from linear programming, for any weight function the maximum weight is taen at one of the extreme points of the convex hull of the profile vectors (the profile polytope) of all families in A We enote the set of profile vectors by µ(a), its convex hull by µ(a), the set of extreme points by E(A) an the families having a profile in E(A), the extremal families by E(A) If the weights are non-negative, then increasing any coorinate of the profile vector increases the weight of the family, so the maximum for these weights is taen at an extreme point which is maximal with respect to the coorinate-wise orering We call these vectors essential extreme points an enote them by E (A) an the corresponing families by E (A) Note that to prove that a set of profiles are the extreme points of the profile polytope one has to express all profiles as a convex combination of these vectors, while to prove that a set of profiles are the essential extreme points of the polytope it is enough to ominate (a vector f ominates g if it is larger in the coorinate-wise orering) any other profiles The systematic investigation of profile vectors an profile polytopes was starte by Erős, Franl an Katona in 4,5, an overview of the topic can be foun in the boo of Engel 3 Corresponing author E-mail aresses: gerbner@cseltehu (D Gerbner), patos@renyihu, tphpab0@ceuhu (B Patós) 00-365X/$ see front matter 008 Elsevier B All rights reserve oi:006/jisc0080707
86 D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 The notion of profile vector can be introuce for any rane poset P (a poset P is sai to be rane if there exist a nonnegative integer l an a mapping r : P {0,,, l} such that for any p, p P if p covers p, we have r(p ) = r(p ) an r(p) = 0 for some p P) In this case the profile of a family F P is efine by f (F ) i = {p F : ran (p) = i} (i = 0,,, n), where ran (p) enotes the ran of an element p an n is the largest ran in P Several results are nown about profile vectors in the generalize context as well (see eg 3,6,) One of the most stuie rane poset is the poset L n (q) of subspaces of an n-imensional vector space over the finite fiel GF(q) with q elements (the orering is just set-theoretic inclusion) In this case the ran of a subspace is just its imension, so the profile vector f (U) of a family U of subspaces is a vector of length n (inexe from 0 to n) with f (U) i = {U U : im U = i}, i = 0,,, n In this paper we etermine the profile polytope of intersecting families in the poset L n (q) A family U of subspaces is calle intersecting if for any U, U U we have im(u U ) (an t- intersecting if for any U, U U we have im(u U ) t) Two subspaces U, U are sai to be isjoint if im(u U ) = 0 ie U U = {0} We will use the symbol q = (qn )(q )(q n ) for the Gaussian (q-nomial) coefficient enoting the number of (q )(q )(q ) -imensional subspaces of an n-imensional space over GF(q) (an q will be omitte, when it is clear from the context) The set of all -imensional subspaces of a vector space will be enote by With the above notations the main result of the present paper is the following theorem Theorem The essential extreme points of the profile polytope of the set of intersecting families of subspaces are the vectors v i ( i n/) for even n an there is an aitional essential extreme point v for o n, where 0 if 0 j i n if i j n i (v i ) j = j () n if j > n i j an 0 if 0 j n/ (v ) j = n if j > n/ j () Intersecting families of subspaces In this section we etermine the essential extreme points of the profile polytope of the set of intersecting families of subspaces (Since the intersecting property is hereitary ie after removing any of its members an intersecting family stays intersecting we now (cf 5) that any extreme point can be obtaine from one of the essential extreme points by changing some of the non-zero coorinates to zero) This was implicitly one in by Bey, but he only state that his results concerning the Boolean lattice stay vali in the context of L n (q) What is more important, our approach is ifferent from his: our main tool in etermining some inequalities concerning the profile vectors of intersecting families of subspaces is Theorem This is a generalization of a theorem of Hsieh 0 which might be of inepenent interest To simplify our counting arguments we introuce the following () Notation If n, then enotes the number of -imensional subspaces of an n-imensional vector space q over GF(q) that are isjoint from a fixe -imensional subspace W of Here are some basic facts about these numbers: Facts () n I = q, II () () (n ) (n ) = q n q an so inuctively for any p III n p p () () q p() (if n) (if n),
D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 863 To etermine the profile polytope of intersecting families we follow the so-calle metho of inequalities Briefly it consists of the following steps: establish as many linear inequalities vali for the profile of any intersecting family as possible (each inequality etermines a halfspace, therefore the profiles must lie in the intersection of all halfspaces etermine by the inequalities), etermine the extreme points of the polytope etermine by the above halfspaces, for all of the above extreme points fin an intersecting family having this extreme point as its profile vector The last step gives that the extreme points of the polytope etermine by the halfspaces are the extreme points of the profile polytope that we are looing for The following theorem on intersecting families was first prove by Hsieh 0 (only for n ) in 977, then by Greene an Kleitman 9 (for the cases n so especially if n = ) in 978 Theorem (Erős Ko Rao for ector Spaces, Hsieh s Theorem) If F then n F is an intersecting family of subspaces an n, The above theorem yiels to the following inequalities concerning the profile vector of any intersecting family: 0 f i, 0 i n/ i 0 f i n i, n/ i n To establish more inequalities we will nee the following statement: Theorem 3 The following generalization of Hsieh s theorem hols: (a) if n an = 0 or = n or (b) if n an n then for any intersecting family F of -imensional subspaces of an n-imensional vector space with all members isjoint from a fixe -imensional subspace U of n () F Note that the = 0 case is just Hsieh s theorem Proof If n or n an = 0 then the argument of Greene an Kleitman 9 wors One can partition \U into isomorphic copies of \{0}, where is a -imensional vector space over GF(q) Since F may contain at most one of the -imensional spaces of each partitioning set, the statement of the theorem follows So now we can assume n We follow the argument in 0 First we verify the valiity of the lemmas from 0 in our context For x (A ) let F x (F A ) enote the set of subspaces in F containing x (A) Lemma A (The Analogue of Lemma 4 in 0) Suppose n an let F be an intersecting family of -subspaces of an n-imensional space such that all -subspaces belonging to F are isjoint from a fixe -imensional subspace W of (where p () p n ) If for all x we have F x, then p n () n () p p F or F A p for all -imensional subspaces A, where p Proof First we chec the valiity of the following consequence of the facts : n () p n () p s p > q p n () p, (3) p p for s p Inee, p p () () p p (n ) (n ) = q p(n ) > q p ( q q ) p p = q p, where the first inequality is Fact III an the secon one uses the assumption n
864 D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 Let us tae an arbitrary -imensional subspace x, y If U F implies U x, y {0}, then by (3) (an the assumption of the lemma) we have p n () p n () p F F Z p p Z x,y,z im Thus we can suppose that there is some U F such that U x, y = {0} Tae 0 z U If U F implies U x, y, z {0}, then (again using (3)) p 3 n () p n () p F p p () n 4 Thus we can suppose that there is some U F such that U x, y, z = {0} Hence F x,y,,z, an so 4 () n 4 F x,y 4 Suppose that for j i, 0 z j U j an x, y, z,, z j U j = {0} Tae 0 z i U i If U F implies U x, y, z,, z i {0}, then by (3) i p 3 n () p n () p F p p Thus we can suppose that there is some U i F such that U i x, y, z,, z i = {0} Hence we have n i () 4 F x,y,z,,z i, i 4 an by inuction we obtain () F x,y i Thus for i p, either we have F p () p F x,y p () or F x,y We will nee one more lemma from Hsieh s paper (actualize to our context): i i i (), as a special case with i = p we have Lemma B (The Analogue of Lemma 43 in 0) Let F be a family of intersecting -subspaces of an n-imensional space of which all subspaces are isjoint from a fixe -imensional subspace W of Furthermore if (a) q 3 an n an for all x we have F x, or if (b) q = an n min{,n } ( an for all x we have F x (n ) empty an equals ), then n () F Proof In all cases F is at most i= times the boun on F x Now if q 3, then ( ) q n (n ) n () F = q q ( )(n ) = q If q =, then for any n an = n we have ( ) ( ) i F (q ) q i= n (n ) = q q ( )(n ) = i ) (if n, then the prouct is
D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 865 Since n, we have n hols This gives ( ) q F = q ( ) (q )(q 3 ) (q ) q (q )(q ) (q ) n (n ) q ( )(n ) = This establishes the lemma for 0 n For the remaining cases (n n ) put a = n (n ) ( ) () i=, b = We have to prove that a b hols for all n n To see this observe that a b a b = = (n ) i () () = q q q q n (n ) q n q = qn q q n q q n q q q (n ) (n ) Thus the sequence a is monotone increasing, an since a n b b n This finishes the proof of the lemma n q ()( ) q ( ) hols, so oes a b for all n n Before we get into the etails of the proof of Theorem 3, we just collect its main ieas: The heart of the proof is the concept of covering number For a family of subsets F n this is the size of the smallest set S n that intersect all sets in F (S nee not be in F ) For a family of subspaces F its covering number is the smallest number τ such that there is a τ -imensional subspace U of that intersects all subspaces that belong to F Observe that the proof of Lemma A was one by an inuction on the covering number The proof of Theorem is again base on an inuction on the covering number of F (During the proof, almost all computations will use the facts about Gaussian coefficients, all inequalities without any further remars follow from them) () If x F for some 0 x then F Thus we can suppose that F = {0} Let x 0 be such that F x = max x F x () n By our assumption, there is some A F not containing x Thus F x Suppose that there are two inepenent vectors z, z A such that A F A x, z i {0} for i =, If u i x, z i \ x, then the u i s are inepenent Thus F F x F U,U U i ( x,z i \ x ) {0},im(U i )= n () ( ) n () n ( ()) Thus we can suppose that there is at most one z A such that A F A x, z {0} Suppose that z A is as such () n 3 Tae x A \ z, then there is some A F such that A x, x = {0} an hence F x,x Thus 3 n () n () 3 F x F x,z F x,x 3 But then F X x,z,im(x)= X (A \ z ) {0},im(X)= ( n ) n () 3 n () F X 3 Thus we can suppose that for all x A there is some A F such that A x, x = {0}, an hence F x,x () n 3 Thus F x 3 n 3 3 ()
866 D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 In general, suppose that for p 3 we have non-zero vectors y, y,, y p an A, A,, A p F such that y i A an A i x, y,, y p = {0} for i p (We have just prove that either for any y A there exists such an A F or the statement of the theorem hols) Thus n p () F x,y,,y p, p an so inuctively we obtain that p n p () F x p By Lemma A, we have p n p () F x,y p for all -imensional x, y Suppose that there are p linearly inepenent vectors z, z,, z p in A p such that x, y,, y p, z i A {0} for all A F an i =,,, p Let u i x, y,, y p, z i \ x, y,, y p, i =,,, p, then u, u,, u p are inepenent Thus F F X F U,U,,U p X x,y,,y p,im(x)= U i ( x,y,,y p,z i \ x,y,,y p ) {0},im(U i )= p p n p () ( ) p p p n p p p p p n p () n p () q (p)( ) p p ( ) p p n p n () p Thus we can suppose that there are at most p such z i s Hence n p () 3 p n p () F x,y,,y p, p 3 p an so p n p () 3 p p n p () F x p 3 p Suppose that we o have inepenent vectors z, z A p such that A F A x, y,, y p, z i {0} for i =, Then F F X F U,U X x,y,,y p,im(x)= U i ( x,y,,y p,z i \ x,y,,y p ) {0},im(U i )= ( p p n p () ) 3 p n p () p 3 p ( ) p p p n p () p p p ( n p () 3 p ) p n p = q (p) p 3 p p p n p () p 3 n p () q p p 3 p p n () n () q p q () ()
D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 867 Thus we can suppose that there is at most one such z Hence p n p () p 3 n p () F x p 3 p Suppose that z A p is such a z, then ( p p n p () p ) 3 n p () F F X p 3 p X x,y,,y p,z,im(x)= p p n p () 3 p n p () p 3 q p p n () n () q p q p Thus we can suppose that for all z A p, there is some A F such A x, y,, y p, z = {0} Tae y p A p, an let A p be such that A x, y,, y p, y p = {0} We obtaine, that either the statement of the theorem hols, or there are linearly inepenent vectors x, y,, y an A i F i =, such that y i A i an x, y, y i A i = {0} Furthermore we can suppose that y i maximizes F x,y,,y i,z for z A i () If q 3, this means that either F or F x F x an then we are one by Lemma B If q =, we have to sharpen our estimates on F x We now that for j inepenent vectors x, y,, y j with U x, y,, y j = {0} there exists a subspace A j F such that A j x, y, x j = {0} Then we woul have () n j F x,y,,y j (Note that U x j, y,, y j = {0} must hol, as otherwise any subspace containing x, y,, y j woul intersect U nontrivially, therefore F x,y,,y j woul be empty, an thus, by the maximality assumption on the choice of y i, F woul be empty) Suppose further that for some positive l we have j = n l Then im( x, y, y j, A j U) l an so (enoting x, y, y j, A j U by U j ) im( x, y, y j, U j A j ) l as well, therefore when choosing among the vectors of A j a subspace of imension at least l is forbien Therefore we have the following better estimate on the number of subspaces in F containing x, y,, y j : ( ) l n j () j Hence we have that either the statement of the theorem hols or the egree of any vector x is boune by the expression given in the conitions of Lemma B So Lemma B establishes our theorem in this case, too Corollary For the profile vector f of any family F of intersecting subspaces of an n-imensional vector space, an for any n/ an n/ n, the following hols c, f f, n where c, = q, an equality hols in the case of f = 0, f = n or f =, f = Proof Let us oublecount the isjoint pairs forme by the elements of F = {U F : im U = } an F = \ F = {U, U F : im U = } On the one han, for each U F there are exactly q such pairs (this uses the first fact about q-nomial coefficients), while on the other han by Theorem 3 we now, that for any W F there are at most () = q ( ) such pairs This proves the require inequality an it is easy to see that equality hols in the n cases state in the Corollary Having establishe these inequalities, we are able to prove our main theorem Proof of Theorem First of all, for any x, for the families G i = {U : x U, i im U n i} {U : im U > n i} ( i n/) f (G i ) = v i hols, an if n is o then the profile of the family G = {U : im U > n/} is v, an clearly none of these vectors can be ominate by any convex combination of the others n
868 D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 We want to ominate the profile vector f of any fixe intersecting family F with a convex combination of the vectors v j (an possibly v if n is o) We efine the coefficients of the v j s recursively Let i enote the inex of the smallest non-zero coorinate of f For all j i let α j = 0 Now if for all j j α j has alreay been efine, let j α j = max f j j j =i α j, 0 Note, that for all j (i j j n/) the jth coorinate of j α =i j v j is at least f j (an equality hols if when choosing α j, the first expression is taen as maximum), so these vectors alreay ominates the first part of f When all α j s (i j n/) are efine, then let α = n/ j α =i j an let α be the coefficient of v if n is o or a α to the coefficient of v n/ if n is even Note also that α is non-negative since for all i j n/ (v j ) = an by Hsieh s theorem 0 f Therefore this is really a convex combination of the v j s The easy observation that this convex combination ominates f in the coorinates larger than n i follows from the fact that all v j s (an v n as well) have in the th coorinate, therefore so oes the convex combination which is clearly an upper boun for f All what remains is to prove the omination in the th coorinates for all n/ n i, that is to prove the inequality n n ( ) n f α j α j Let n be the largest inex with α > 0 Then we have ( n f c, f = c, α j = ( ) n n n = α j α j ) α j n α j where the inequality is just the Corollary, the first equality follows from the fact that α > 0, the secon equality uses again the Corollary (the statement about when equality hols) an the last equality uses the efining property of (for all j n α j = 0) This proves the theorem Note that, the (essential) extreme points are the same as in the Boolean case, one just has to change the binomial coefficients to the corresponing q-nomial coefficients an the structures of the extremal families are really the same 3 Concluing remars The authors of this paper in 8 introuce a generalization of the notion of profile vector, the so-calle l-chain profile vector, where the coorinates are inexe by j, j,, j l (0 j j j l ran (P)) an count the number of chains of length l in the family where the ith element of the chain shoul have ran j i for all i l (so the l-chain ( profile vector of a family has n l ) coorinates in the Boolean poset an in L n (q) as well) As the set of intersecting family is upwar close (ie if F is an intersecting family of subspaces of, then so is U(F ) = {W : U F (U )}), one can obtain the essential extreme points of the l-chain profile polytope for any l as escribe in 8 The profile polytope of t-intersecting families has not yet been etermine neither in the Boolean case nor in the poset of subspaces, but in both cases we now how large can be the ith coorinate of the profile for all 0 i n Theorem 4 (Franl Wilson 7) If U U max { t t, t } is a t-intersecting family an n t, then The corresponing extremal families are i U 0 = {U : T U} where T is a fixe t-imensional subspace of, ii U = where W is a fixe t-imensional subspace of W
Theorem 5 (Ahlswee Khachatrian ) If t n an F D Gerbner, B Patós / Discrete Mathematics 309 (009) 86 869 869 F max F r, 0r n t ( ) where F r = {F n : F, t r t i} for 0 r n t ( ) n is a t-intersecting family, then These two theorems show that in the case of subspaces the extremal family is always one of two caniates, while in the Boolean case (as n goes to infinity) there are arbitrary many caniates (in fact for all r Theorem 5 in its full strength gives the range of where F r is the extremal family) Therefore one may suspect that it can be much easier to etermine the profile polytope in the lattice of subspaces, than etermining it in the Boolean case Acnowlegements We woul lie to than Péter L Erős an Gyula OH Katona for their useful comments on the first version of the manuscript We woul also lie to than the anonymous referees for the careful reaing an for calling our attention to the paper of Bey References R Ahlswee, L Khachatrian, The complete intersection theorem for systems of finite sets, European J Combin 8 (997) 5 36 C Bey, Polynomial LYM inequalities, Combinatorica 5 (005) 9 38 3 K Engel, Sperner Theory, in: Encyclopeia of Mathematics an its Applications, vol 65, Cambrige University Press, Cambrige, 997, p x47 4 PL Erős, P Franl, GOH Katona, Intersecting Sperner families an their convex hulls, Combinatorica 4 (984) 34 5 PL Erős, P Franl, GOH Katona, Extremal hypergraphs problems an convex hulls, Combinatorica 5 (985) 6 6 P Franl, The convex hull of antichains in posets, Combinatorica (4) (99) 493 496 7 P Franl, RM Wilson, The Erős-Ko-Rao theorem for vector spaces, J Combin Theory Ser A 43 () (986) 8 36 8 D Gerbner, B Patós, l-chain profile vectors, SIAM J Discrete Math () (008) 85 93 9 C Greene, DJ Kleitman, Proof techniques in the theory of finite sets, in: Stuies in Combinatorics, in: MAA Stu Math, vol 7, Math Assoc America, Washington, DC, 978, pp 79 0 WN Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math (975) 6 A Sali, A note on convex hulls of more-part Sperner families, J Combin Theory Ser A 49 (988) 88 90