IAI = r} given by A < B if max (A t:. B) E B. The Kruskal-Katona theorem ('7).

Transcription

1 Eurp. J. Cmbinatrics (1990) 11, Isperimetric Inequalities fr Faces f the Cube and the Grid B. BOLLOBAs AND A. J. RADCLIFFE A face f the cube rj>(n) = {O, l} N is a subset determined by fixing the values f sme crdinates and allwing the remainder free rein. Fr instance, the edges f the cube are faces f dimensin 1. In Sectin 2 f this paper we prve a best pssible upper bund fr the number f i-faces f rj>(n) cntained in any subset f rj>(n). In particular, we shw that initial segments in the binary rdering-the rdering n rj>(n) induced by the map A... E i e A 2;: rj>(n) --+ Ncntain the greatest pssible number f i-faces fr any i ;;,. O. In Sectin 3 the inequality is extended t apply t the grid [p t fr p;;" 2, and t give a bund n the number f i-dimensinal faces enclsed by a cllectin f j-dimensinal faces, fr i ;;,. j. Finally, in Sectin 4, we apply the face isperimetric result t the prblem which riginally mtivated its study. We prve a Kruskal-Katna type result fr dwn-sets in the grid. 1. INTRODUCTION Suppse we are given a set system d, i.e. a subset f ~ ( = N ) ~ ( { 2, 1...,,N}) = {O, 1}N, which is a dwn-set: whenever A E d and B ca then BEd. If d cntains a certain number f sets f size r (i.e. r-setsi at least hw large is d? This questin is neatly and definitively answered by the Kruskal-Katna therem ([9], [8]). The lwer shadw f a cllectin 91l f r-sets is the cllectin f all (r - Ij-sets cntained in sme member f 9/l. The Kruskal-Katna therem gives a best pssible lwer bund fr the size.f the lwer shadw f a cllectin f r-sets, given its size. In fact, the minimum is attained by initial segments in the clex rdering, the rdering n [N](r) = {A E IAI = r} given by A < B if max (A t:. B) E B. The Kruskal-Katna therem ('7). ~ ( N ) : easily implies that if d n [N](r) has size at least (':.') fr sme m, then Idl;;.: ~ ~ Furthermre, as was shwn by Lvasz ([10], 13 Prblem 31), the same hlds with m nt an integer. Mtivated by prblems arising in the gemetric study f cnvex bdies, we cnsider here the same prblem in the grid [p]n = {O, 1,...,p -1}N. The grid is a lattice in a natural way s the ntin f a dwn-set is still defined: a subset S c [p]n such that fr any sequence s belnging t S all sequences pintwise smaller than s als belng. We take the weight f a sequence s E [p ]N t be the number f crdinates in which it takes the value p - 1. If a dwn-set in [p]n cntains a certain number f sequences f weight r, hw large must it be? The prblem turns ut t be equivalent t an isperimetric prblem fr the faces f the grid. We first slve this prblem, and nly return t the generalized Kruskal-Katna therem in Sectin 4. In general an isperimetric inequality is a relatin between the size f a set and the size f its bundary. Fr instance the classical isperimetric inequality in IR n states that any measurable subset f IR n f measure I.l has bundary measure at least as great as that f the spherical ball f measure I.l. Of curse in different situatins we must chse different definitins fr the bundary f a set. If we cnsider subsets S f a graph G then tw natural chices are the vertex bundary, the number f vertices in G - S adjacent t vertices f S, and the edge bundary, the number f edges jining S t G - S. In a bipartite graph H, with bipartitin (V t, V 2 ), the (vertex) bundary f a subset S c VI is the cllectin r(s) f its neighburs in V 2 (Nte that the crrespnding ntin f edge bundary is uninteresting: the size f the edge bundary f a subset /90/ $02.00/ Academic Press Limited

2 324 B. Bllbtis and A. J. Radcliffe S C Vi is merely the sum f the degrees f its elements.) The isperimetric prblems cnsidered here are cncerned with this last ntin. The aim is t minimize Ir(S)1 amng subsets S c Vi f fixed size. There are many examples f isperimetric results. Harper [6J gave a slutin t the vertex isperimetric prblem in the cube, i.e. in the graph n rfj(n) in which tw sets A and B are cnnected if IA 6 BI = 1. Harper, Bernstein and Hart ([5], [2J, [7]) gave a best pssible edge isperimetric inequality in the same graph. Bth these therems, and several ther isperimetric results, are presented in [3J. Aln and Milman [IJ and Bllbas and Leader [4J have each, by widely different methds, prved isperimetric inequalities valid fr subsets f arbitrary prducts f cnnected graphs. The Kruskal-Katna therem ([9], [8]) is als an isperimetric result. Define a bipartite graph n [Nt) U [NJ(r-i) by jining A E [Nt) t BE [Nt-I) when B ca. The Kruskal-Katna therem slves the isperimetric prblem fr this graph. The results we prve are isperimetric results fr the faces f the cube rfj(n). In all f them the parameter N is essentially irrelevant; we culd replace N by N and ur results (and prfs) wuld be unaffected. Hwever, the results are f a finite nature. Fr a set Dc [NJ we write DC fr {I, 2,..., N} - D, the cmplement f Din [NJ. A face f rfj(n) is a subset f the frm F B C = {A E rfj(n): An W = C} fr sme Be {I, 2,..., N} and C c B C In ther wrds, it is a subset btained by fixing the values f sme crdinates and allwing the remainder free rein. The dimensin f FB,c is IBI, s the faces f dimensin 1 are exactly the edges f rfj(n). Thus there are 2(N - i)('f) faces f dimens in i, crrespnding t the ('f) pssibilities fr B and the 2 N - i chices f C c B C We culd equally call faces sub-cubes. FB,c is a translated cpy f rfj(b) based at C. We dente by ~the cllectin f all the i-dimensinal faces f rfj(n). If i ~j ~0 we define a bipartite graph Bi,i as fllws. It has bipartitin (, ~ ~ and ) we jin tw faces, Fj E ~and F2 E ~ if, F2 c Fj. In Sectin 2 we give a best pssible isperimetric inequality fr this bipartite graph when j = O. This generalizes the edge isperimetric inequality f Harper, Bernstein and Hart ([5J, [2], [7]), which is the case i = 1, j = O. In Sectin 3 we extend the face isperimetric result t the grid [p IN. Mrever, the extensin applies t faces f any pair f dimensins. Fr the results f this sectin we replace ur previus exact estimates by smth apprximatins, which give best pssible answers in the mst imprtant cases. Finally, in Sectin 4, we apply the face isperimetric result in the grid t give a new generalizatin f the Kruskal-Katna therem. 2. AN ISOPERIMETRIC INEQUALITY FOR THE FACES OF A CUBE In this sectin we shall study the bipartite graph Bi, frmed by the 2 N - i ('f) i-dimensinal faces and the 2 N O-dimensinal faces, i.e. vertices. T slve the isperimetric prblem in Bi, we need t determine, fr all k ~1, min{!f(d)l:.91 c, ~ = k}. Equivalently, it suffices t determine, fr all k ~1, max{ldl:!f(d)1 ~k,.91 c ~}. The prblem then becmes ne f finding the maximum number f i-dimensinal faces that can be enclsed by k vertices. This maximum, we shall prve, is exactly the number f i-faces spanned by the initial segment f length k in the binary rder: the rdering n rfj(n) induced by the map A ~I: i e A 2 i : rfj(n)- N. Harper, Bernstein and Hart ([5], [2], [7]) prved this when i = 1 and Hart's prf was simplified slightly in [3J.

3 Isperimetric inequalities 325 Fllwing this last prf we btain an inequality abut the extremal functin and then deduce the isperimetric result. Mst f the wrk ges int prving the inequality, here presented as fllws. Let i, j, I, m be psitive integers. Define h(j) t be the number f Is in the binary representatin f j. Then set Similarly, if I c r\l let t(/, m) = ~ 1( h ~ j» ). j=t l Nte that t(/, m) =t([l, m» and 1(I) = Ill- Our extremal functin will be t(o, m), which is exactly the number f i-dimensinal faces cntained in the first m elements f ~ ( in N the ) binary rder. This is clear if ne recalls that the first m binary numbers are the characteristic vectrs f this initial segment, and als ntes that the initial segment, since it is a dwn-set, cntains a face FB,c whenever it cntains B U C. In ur prfs we ften want t 'shift' an interval by sme pwer f tw. If I is an interval in r\l which starts later than 2' and is shrter than 2', it is clear frm the definitin that fr all i;:;. 1, The next few lemmas prvide basic infrmatin abut the extremal functin. A cyclic sub-interval f [0, 2') is a subset either f the frm [I, m) n [0, 2') r f the frm [0, 2') - [I, m) fr sme I, m e [0, 2'). LEMMA 1. Letk, r e r\l be such that k ~2' and suppse that I is a cyclic sub-interval f [0, 2') f length k. Then t(i);:;.t(o, k). PROOF. Write A(j) fr the set f crdinates in which the binary expansin f j has a ne, s that IA(j)1 = h(j). Nting that t(i) =L ( h ~ = j ) L) I{j E I: A ca(j)}i, jet l AeN(') it certainly suffices t prve that fr all A e r\l(i) we have I{j E I: A c A (j)}i at mst l{je[o, k):aca(j)}i. We prve this by inductin n i, nting that it is trivial fr i =0. Cnsider any i-set A and write JA fr the set f all j in r\l with A c A(j). The characteristic functin f J A is peridic with perid 2 max (A ) + \ and thus we need t prve that all (cyclic) sub-intervals f [0, 2 max (A ) + 1) f length k have larger intersectin with J A than des [0, k). Let I be such an interval. If k ~ 2 m a) then x ( A[0, k) nj A is empty and there is nthing t prve. If k > 2 max (A ) then II n JAI = II n [2 max (A ), 2 m ax (A ) + 1) n JA-{max(A )}I ;:;'1[0, k - 2 max (A» nja-{max(a )}1 = 1[0, k)njai The inequality hlds because II n [2 max (A ), 2 max (A ) + 1)I;:;. k - 2 max (A ) and because, by inductin, [0, k - 2 m ax (A» has the smallest intersectin with JA-{max (A )} amng intervals f length k - 2 max (A ). 0

4 326 B. Bllbds and A. J. Radcliffe Lemma 1 states that t minimize /;(1) it is best t chse I at the very beginning f ~.By mimicking its prf ne can shw that /;(1) is maximized at the end f binary intervals. LEMMA 2. Let k, r E ~be such that k ~2 r and suppse that I is a cyclic sub-interval f length kin [0, 2 r). Then /;(1) ~ / ; -( k, 2 r2 r). Using Lemmas 1 and 2 we prve the fllwing tw cmplementary lemmas, which are special cases f the inequality t be prved in Therem 5. LEMMA 3. Fr all i ~0, if k, I, r E ~satisfy k ~2 r ~I then PROOF. /;(/, 1+ k) ~/;(2 r, 2 r + k). Fr all j in [I, 1+ k) there is sme element f AU) f size at least r. S if the interval [I, I + k) is replaced by the (cyclic) interval I = [I md 2 r, (I + k) md 2 r ) c [0, 2 r ), we have, applying Lemma 1, /;(1, 1+ k) ~/;(1) +fi-1(1) ~/;(0, k) +fi-1(0, k) =/;(2 r, 2 r + k). LEMMA 4. Fr all i ~0, if k, I, r E ~satisfy 1+ k ~2 r- 1 then /;(/, 1+ k) +f i-l(1, 1+ k) ~ / ; ( 2 r 2-r k)., Als, if k, I, r E ~satisfy I ~2 r and 1+ k ~2 r + 2 r- 1 then PROOF. /;(/, 1+ k) ~ / ; ( 2 r 2-r k)., Fr the first inequality, mimic the prf f Lemma 3, using Lemma 2 in place f Lemma 1. Nw if k, I, r are such that I ~2 r and 1+ k ~2 r + 2 r- 1 then we have, by the first inequality, /;(1, I+ k) =/;(1-2 r, I+k - 2 r) +/;-1(1-2 r, 1+ k - 2 r) ~ / ; ( 22 r -r k)., We are nw in a psitin t prve the crucial inequality n which ur isperimetric therem rests. THEOREM 5. Fr all i ~0, if k, I, r E N satisfy k ~I and k ~2 r then /;(/, 1+ k) ~/;(2 r, 2 r + k). Als if k, I, r satisfy 2k + I ~2 r + 1 and I ~2 r (which imply k ~2 r- 1) we have /;(1, I + k) ~/;(2 r - k, 2 r ). (1) (2) PROOF. We prve (1) and (2) by inductin n k + r, at each stage assuming that bth (1) and (2) are true fr smaller values. Turning first t (1) we see that Lemma 3 is exactly (1) in the case I ~2 r, s we may assume 1< 2 r. Again, if 1+ k ~2 r then k ~2 r - 1 and by inductin we knw /;(/, 1+ k) ~/;(2 r - 1, 2 r- 1 + k) =/;(2 r, 2 r + k).

5 Isperimetric inequalities 327 In the nly remaining situatin, 2' 'splits' the interval [I, 1+ k), i.e. 1<2' < I + k. Set I' = I + k, k' = 2' -I, r' = r. Then 2k' + I' = 2,+1 -I + k ~2"+1, l ' = I + k and ~ 2 " k'<k, s we may apply the inductin hypthesis; t be precise (2) implies that Thus /;(1 + k, 2' + k) =/;(1', I' + k') ~ / ; -( k', 2 " 2") =/;(/,2'). /;(/, 1+ k) = /;(1,2') + /;(2', 1+ k) ~/;(1 + k, 2' + k) + /;(2', 1+ k) =/;(2',2' + k), and (1) is prved. Let us nw turn t (2). If 1+ k ~2' + 2,-1 then the secnd part f Lemma 4 is exactly what is required. Otherwise, (2) is equivalent t /;(1-2', 1+ k - 2') + /;-1(/- 2', 1+ k - 2') ~ / ; -( k, 2 ' 2'). (3) If1-2' ~2'-1 then, setting I' = 1-2', k' = k and r' = r - 1 in (2), apply the inductin hypthesis twice, nce t /; and nce t 1;-1> t give and /;(1-2', I + k - 2') = /;(/', I' + k') ~/;(2" - k', 2'') = /;(2,-1 - k, 2'-1) /;-1(/- 2', 1+ k - 2') = 1;-1(/', I' + k') ~ / ; - -1 k', ( 22'') " = /;_1(2,-1 - k, 2,-1). Summing these inequalities we btain (3). We are nce again in the situatin where the interval f interest, in this case [1-2', 1+ k - 2'), is 'split': 1-2' ~2,-1 ~I + k - 2'. Since /;(1-2',2'-1) + 1;-1(/ 2',2'-1) =/;(2' - k, 1-2'), it nly remains t shw that /;(2,-1, 1+ k - 2') + /; _1(2'-1, 1+ k - 2') ~ / ; -( k, 2 ' 1-2'). (4) The left-hand side is /;(2', 1+ k - 2,-1) and s (4) is an instance f (1), with I' = 2' - k, k' =I + k - (2' + 2'-1) and r' =r - 1. We need nly check that k' ~l' and k' ~2" t shw that by inductin (4) hlds. These inequalities are true since and I' - k' = 2' + 3.2'-1_1-2k =5.2'-1- (2k + I) k' = I + k - 3.2'-1 This cmpletes the prf f Therem 5. ~2'-1. ~ O The face isperimetric result is, as remarked earlier, a crllary f Therem 5. Given a set system de PJ>(N), write b;(d) fr the number f i-dimensinal faces enclsed by d. THEOREM 6. Let i E N and let de PJ>(N) be a set system. Then b;(d) ~ / ; Idl). ( O,

6 328 B. Bllbtis and A. J. Radcliffe PROOF. Again we use inductin n i. If i = 0 the result is bvius. Otherwise, split d int tw parts accrding as 1 is r is nt a member: d 1 = {A Ed: 1 EA}, d = d - d I- Withut lss f generality bth parts are strictly smaller than d. Set k =Idl, k= Idl and k, = Idd, and suppse that k, ~k. The faces f!jp(n) spanned by d are either spanned cmpletely by ne f d and d 1 r split equally between the tw. Thus bi(d) ~bi(d ) + bi(d 1 ) + bi-1(d1) ~ / ; k) ( O +/;(0,, k 1 ) + fi-l(o, k 1 ) = /;(0, k) - /;(k, k) +/;(0, k 1 ) + fi-l(o, k 1 ) ~ / ; k) ( O =/;(0,, Idl). The final inequality is an applictin f Therem EXTENSIONS TO THE GRID When dealing with the grid [pt, the natural generalizatin f a face is a cllectin btained by fixing the values f sme crdinates and leaving the rest arbitrary. Thus we define an i-dimensinal face f [p]n t be a subset f the frm FB,t = {s E [pt:slbc= t} fr sme Be. {1, 2,...,N} with IBI = i and sme t E [p]bc. There are pn-i("f) faces f dimensin i. A subset d f [p]n enclses a face F if Fe. d. If d is a cllectin f i-dimensinal faces f [pt we say it enclses a j-face F if all the i-faces cntained in F are members f d. In either case we write bj,p(d) fr the number f j-faces enclsed by d. Let fp( dente the cllectin f all i-faces f [p ]N. Again, whenever i x ~i -1; x ~ i - 1. ~j, define a bipartite graph BL n fp( U 'iff by jining Pi E fp( t F2 E '!F'j if F2 c. Pi. Fr j > 0 r p > 2 the cnjectured extremal functin fr the isperimetric prblem in Bfj (see Cnjecture 12) becmes unwieldy. Instead f dealing with this cmplicated extremal functin we use a smth apprximatin t btain an isperimetric result, which is best pssible in many places and never severely in errr. T give the flavur f the results, we first prve a generalizatin f Therem 6 t the grid, again by establishing an inequality fr the bunding functin. Afterwards, we prve the mre technical Lemma 9, which in turn gives Therem 10, the face isperimetric result in its full generality. Recall that we define (D fr all x E ~and i E N by e) = x(x - 1)... (x - i + 1), {., ' l. Define P;(x) t be i! (D, and fr cnvenience let P be the cnstant functin 1 and let P- 1 be the cnstant functin O. Nw, if i E Nand p ~2, define gi,p(x) = x c X)/ ~ pi, p The fllwing inequality fr the functin gi,p is the basis fr the first face isperimetric inequality in the grid. LEMMA 7. Let i ~1, P ~2. If (X/)f':OI E ~ satisfies p min{xj} =X ~«:'. then g i' P Xj) ( ~ ~gi-l,p(xo) + : gi,p(xj)' ~ (5)

7 lsperimetric inequalities 329 PROOF. Set m = X and Y; =x; - X fr j E [p] and multiply bth sides f (5) by p'ii. With this substitutin the cnditins becme 0:;;; m, Yl'..., Yp-l and the inequality t be prved becmes: (pm + :t: Y; )p;(lg,(pm + :t: Y;)) p-l ~ipmpi-i(lgp m) + 2: (m + yjp;(lg,(m +Yj))' (6) ;=0 Nw if Yl = Y2 =... = Yp-l = 0 then we have equality in (6), fr P;(Igppm) = P;(lgp m + 1) = ipi-1(lo&> m) + Pi(lgp m). Als nte that, since bth P; and P; are increasing functins, 1 Y;)P;(lgp(pm +Pi Y;))) ~((pm + ~ : f 0Yk ] =1 ]=1 =P;(lgp(pm +:t:y;)) + P;(lgp (pm +:t:yj)) / I O ~ p ~p;(lgp(m +Yj)) + P;(lgp(m + yj)/lgep p =- 2: (m + y;)p;(lgp(m +Yj))' Ykj=l fr any k E {1,..., p - 1}. Since (6) is true when YI =... = Yp-l = 0 and the partial derivatives f the left-hand side are always greater than the crrespnding partial derivatives f the right, (6) is prved. 0 The preceding estimate n gi,p is sufficient t prve the ur first face isperimetric result fr the grid, an analgue f Therem 6 fr the cube. THEOREM 8. Let i E N, p ~2, and let stj be a finite subset f [pt. Then PROOF. bi,p(stj):;;; gi,p(istji). We prve the result by inductin n i. If i =0 the result is bvius, s suppse i ~1. Partitin stj int p parts by the value f the first crdinate: stj; = {s E stj: Sl = n. Set x = IstJl and Xj = IstJjl. Withut lss f generality we may suppse each f the Xi strictly less than k and that X is the smallest f the x;. The i-faces spanned by stj are either entirely cntained within ne f the stj; r are split evenly acrss the varius layers. These latter can ttal n mre than the number f (i - 1)-faces f stj, since each is uniquely specified by its intersectin with stj. S p-l bi,p(stj):;;; bi-i,p(stjo) + 2: bi,p(stj;) j=o p-l :;;; gi-l;p(xo) + 2: gi,p(x;) ; =0 :;;; g i, p X;) ( ~ = i lgi,p(istjl). ]=0 The last inequality is easily seen t hld fr X < pi-i, as then bi-i,p(stj) = 0, and fr X O ~ p i - l it is precisely Lemma 7. T prve an isperimetric result in Hf.j valid fr all p ~2 and all i ~j, we need an estimate fr gi,p which is mre delicate than that in Lemma 7.

8 330 B. Bllbds and A. J. Radcliffe LEMMA 9. Suppse i ~ O P, ~ and 2 m, Y, Yl'".,Yp-l, X E ~ + If. x is the largest slutin f gi,p(x)=gi-l,p(m) + ~ b gi,p(m - l +Yj), s x =pi-l when the right-hand side is 0, then p-l gi+l,p(x) ~gi,p(m) + L gi+l,p(m +Yj)' (7) PROOf. Cnsider first the case where Yl =Y2 =... =Y P = O. If m ~pi-2 then the right-hand side f (7) is zer. If, n the ther hand, m equality in (7) since gi+l,p(pm) =gi,p(m) + pgi+l,p(m). If pi-2 < m < pi-l then a simple calculatin shws that (7) hlds, As abve, we differentiate bth sides f (7). Fix k E [p]. Nting that we have On the ther hand, ax ayk g;'p(m +Yj) g;,p(x) - = = ~ :. ~ a, ax g;+l,p(x), -;- gi+l.p(x) = gi+l,p(x) -;-=, ( ) gi,p(m +Yj)' dyk dyk gi.p X -::.a dyk Thus it suffices t prve that (gi.p(m) +i gi+l.p(m +Yj») = g;+l,p(m +Yj)' j=l ~pi-l then x = pm and we have g;,p(x)g;+l,p(m + yj ~ g ; + l, p ( x +Yj)' ) g ; ' p ( m (8) If m +Yj ~pi then the left-hand side is zer and there is nthing t prve; we may assume in the remainder f the prf that m +Yj ~pi. Equatin (8) claims that in this regin g;+l.p/g;'p is an increasing functin. T prve this, examine the sign f its derivative, which is the same as the sign f g7+1,pg;,p - g ; + l, ~ we ' / want. p ; t prve that Clearly, and g7+1,p(x)g;,p(x) ~g;+ l,p(x)g'/.p(x). (9) ~ ( ) = _1_ (P(l ) P;(lgp X») g,p x 'f i,gp X + I ' lp ~ p '! ( ) = _1_ (P;(lgp x) P7(lgp X») g,p x i!p' x l ~ px ( l ~ p ) Multiplying thrugh by i! (i + 1)! p2i+lx lg, p, (9) reduces t the fllwing: ( P;+l()') + P7+1(A») (P;(A)+ P;(A») ~(Pi+1(A) + P;+l(A») (P;(A) + P7(A»). (10) lg.p lg.p l ~ p l ~ p Nw use the fact that P;+l(A) = (A - i)p;(a), s P;(A) = P;(A) + (A - i)p;(a) and P7(A) = 2P;(A) + (A- i)p7(a). Thus and P;(A)P;+l(A) = P;(A)(P;(A) + (A - i)p;(a» ~P;+l(A)P;(A)

9 Isperimetric inequalities 331 Again we can reduce (t), cancelling terms and using the tw inequalities abve. What remains t be shwn that Since and P;+1(A)Pi(A) = (P;(A) + (A - i)p;(a»pi(a), it suffices t shw that This is straightfrward frm the definitin f P;(A). Indeed, simply nte that P;(A) = P;(A) ( i ) A- k ' ~ and (11) is clear, even withut the factr f 2 n the left. (11) Nw we can prve the main result f the paper, an isperimetric inequality valid fr all grids and between any pair f levels. THEOREM 10. Let 00:::; i 0:::; k be integers. Let p ;;:. 2 and suppse that st/ is a cllectin fi-dimensinal faces f [pt fsize gi,p(x), with x ;;:. pi-1. Then PROOF, bk,p(st/)0:::; gk,p(x), It clearly suffices t prve the result fr k = i + 1, and this we d by inductin n N + i. Partitin st/ int p + 1 parts; p parts cnsisting f the i-faces which are at a fixed height, i.e. have a cnstant value fr the first crdinate, partitined accrding t their height, and ne part cnsisting f the remainder, the i-faces in st/ which are split evenly between the varius hrizntal layers. In ther wrds, set d j = {F e st/: S1 =j fr all s e F} and Ai = st/- U=l d j. Let m, X, Xl>, Xp-1 satisfy I.MI =gi-1,p(m) and Ist/jl =gi,p(xj)' Certainly, p-1 gi,p(x) = gi-1,p(m) + L gi,p(xj)' j=o The (i + Ij-faces enclsed by st/ fall int tw classes, thse entirely cntained within ne layer and thse split between all the layers. Write Y and Y' fr these tw classes, The latter are cnstrained in tw ways; bth their 'hrizntal' faces and their 'vertical' faces must be members f d. The first cnditin ensures that there are n mre than minje[p) 1st/A = minje[p) gi,p(xj) f them. T see the effect f the secnd, nte that these vertical faces are determined by their intersectin with any hrizntal layer. The cnditin is that the cllectin {F n [pt- 1 : Fe.M} must span each face in {F n [pt- 1:FeY'}. (Here we write [p]n-1 fr {se[pt:s1=0}). Thus, by inductin, W'I 0:::; bi,p(.m) :::;gi,p(m). This gives us the fllwing relatins: p-1 bi+1,p(st/) = WI + W'I 0:::; min{gi,p(m), g i. p ( X j + ) L} f bi+1,p(dj) ~ i l j=o p-1 0:::; min{gi,p (m), g i. p ( X+ j L) } gi+1,p(xj)' f ~ j=o

10 332 B. Bllbas and A. J. Radcliffe Set m' = min{m, Xj}f':l and let x' satisfy gi.p(x') =gi-l,p(m') + ~ f ' : l m' we certainly have x' ~ X Using. Lemma 9 we find that ~ m p-l bi+1,p(d) ~ g i, p ( m ' ) This calculatin finishes the prf f Therem L gi+l,p(xj) j=o ~ g i + l, ~ p g( i X + ' l ), p ( X ). gi,p(xj)' Since If ne takes x = py in the previus therem, fr sme y integer, ne btains the fllwing result. ~i, nt necessarily an COROLLARY 11. Let 0 ~i ~k be integers. Let p ~2 and suppse that d is a cllectin fi-dimensinal faces f [p]n fsize py-i(n, fr sme y ~i. Then the number f k-dimensinal faces spanned by d is at mst py-k(k). 0 T cnclude this nte, we remark that althugh when y is an integer Crllary 11 is best pssible, the questin remains pen f whether, as in the edge isperimetric inequality, there is sme rdering whse initial segments are always best. In rder t state a cnjecture n this prblem we intrduce the facial rder. This is the rdering n [pt, defined as fllws: s -ct if S-l(j) cmes after r1(j) in the binary rder, where j is the smallest element f [p] fr which these tw differ. The first few elements in the facial rder fr p =3, are, suppressing trailing zeres: 0, 1, 2, 01, 02, 11, 21, 12, 22, 001, 002, 101, 201, 102, 202,... Thus fr a sequence t be early in the facial rder the first pririty is that the set f zeres shuld be late in the binary rder. After that, the mst imprtant cnsideratin is that the set f Is shuld be late; after that, that the set f 2s shuld be late, and s n. Anther way t regard the rdering is as fllws. Suppse we have written dwn the initial segment f length 3 i, split int varius 'blcks'. In rder t write dwn the next 2.3 i sequences we take each blck in turn and write it twice, nce with a 1 in crdinate i + 1, nce with a 2 in crdinate i + 1. These tw blcks tgether then frm a new blck. S first we write dwn the all zer sequence, a blck n its wn. This is fllwed by the sequences 1 and 2 (using the same cnventin f suppressing trailing zeres) frming a blck tgether. S far we have reached three sequences, i.e. i = 1. The next blck is a dubling f the very first: 01,02. Then cmes a blck f fur arising frm the secnd and third sequences: 11, 21, 12, 22. We have reached nine sequences and the cycle restarts: 001, 002; 101, 201, 102, 202; 011, 021, 012, 022;... We can nw state ur cnjecture. CONJECTURE 12. If d is a subset f [p]n f size k and ~ p ( isk the ) initial segment f size k in the facial rder, then fr all i ~0 the number f i-faces spanned by ~ p ( isk at) least as great as the number spanned by d A KRUSKAL-KATONA THEOREM FOR THE GRID We nw return t the questin psed in the intrductin. Recall that the weight f a sequence s E [ptis Is-1(p -1)1, in ther wrds the number f crdinates in which s take the value p - 1, and let L c ~ [p]n dente the cllectin f all sequences f weight r. The questin is: if a dwn-set de [p]n has Idn L ~ =k, I what can we say abut Idl? This prblem is a disguised versin f the face isperimetric prblem.

11 lsperimetric inequalities 333 We set up a crrespndence between L ~and ; g ; ~ when, r> 3, as fllws: given a sequence s f weight r, it crrespnds t the set f sequences F S btainable by replacing all the ccurrences f p - 1 in s with elements f [p - 1]. In ther wrds, if B = S-I(p -1) and s' = slbe then s crrespnds t FB. s " Crdinates where s takes the value p - 1 crrespnd t 'free' crdinates in the crrespnding face. If the sequence t is the same as s except that in sme crdinates t takes a value in [p -1] where s takes the value p - 1, then F' c PS, Therefre, under this identificatin, the graph Bfjl is ismrphic t the bipartite graph n LfU Lf, where s e Ls is jined t t E Lf if s.>t i fr all i E {I, 2,..., N}. Thus the fllwing tw results fllw immediately frm Therem 10. THEOREM 13. If p;;;'3 and.sil is a subset f L ~f size (p -ly-rg) fr sme x;;;' r -1, nt necessarily an integer, then the number f sequences in L ~ _ which I are pintwise smaller than sme sequence in.sil is at least (p - 1y-r+l(r': I)' D THEOREM 14. satisfies 1.sIl n L ~ I ; ; ; ' als Suppse that p ;;;. 3 and that.sil is a dwn-set in the grid [p]n which (p - 1y-rG) fr sme x;;;' r -1, nt necessarily an integer. Then 1.sIli ;;;.±(p - 1 Y- j ( ~ ). j =O J D REFERENCES 1. N. Aln and V. Milman, A\, isperimetric inequalities and supercncentratrs, J. Cmbin. Thery (B), 38 (1985), A. J. Bernstein, Maximally cnnected arrays n the n-eube, SIAM J. Appl. Math., 15 (1967), B. Bllbas, Cmbinatrics: Set Systems, Hypergraphs, Families f Vectrs and Cmbinatrial Prbability, Cambridge University Press, Cambridge, 1986, xii + 177pp, 4. B. Bllbas and I. Leader, Cmpressins and isperimetric inequalities, t appear in J. Cmbin. Thery (A). 5. L. H. Harper, Optimal assignments f numbers t vertices, J. Sc. Indust, Appl. Math., 12 (1964), L. H. Harper, Optimal numberings and isperimetric prblems n graphs, J. Cmbin. Thery, 1 (1966), S. Hart, On the number f edges f the n-cube, Discr. Math., 14 (1976), G. O. H. Katna, A therem n finite sets, in: Thery fgraphs (P. Erds and G. O. H. Katna, eds), Akademiai Kiad6, Budapest, 1968, pp J. B. Kruskal, The number f simplices in a cmplex, in: Mathematical Optimizatin Techniques, University f Califrnia Press, Berkeley, 1963, pp L. Lvasz, Cmbinatrial Prblems and Exercises, Nrth-Hlland, Amsterdam-New Yrk-Oxfrd, 1979, 551 pp. Received 25 February 1989 and accepted 12 February 1990 B. BOLLOBAS AND A. J. RADCLIFFE Department f Pure Mathematics and Mathematical Statistics, University f Cambridge. 16 Mill Lane, Cambridge CB21SB, U.K. and Department fmathematics, Luisiana State University, Batn Ruge, LA 70803, U.S.A.