Dedicated to the memory of Lev Meshalkin.

Transcription

1 A Meshalki Theorem for Projective Geometries Matthias Beck ad Thomas Zaslavsky 2 Departmet of Mathematical Scieces State Uiversity of New York at Bighamto Bighamto, NY, U.S.A matthias@math.bighamto.edu zaslav@math.bighamto.edu Versio of February 5, Dedicated to the memory of Lev Meshalki. Abstract: Let M be a family of seueces (a,..., a p ) where each a k is a flat i a projective geometry of rak (dimesio ) ad order, ad the sum of raks, r(a ) + + r(a p ), euals the rak of the joi a a p. We prove upper bouds o M ad correspodig LYM ieualities assumig that (i) all jois are the whole geometry ad for each k < p the set of all a k s of seueces i M cotais o chai of legth l, ad that (ii) the jois are arbitrary ad the chai coditio holds for all k. These results are -aalogs of geeralizatios of Meshalki s ad Erdős s geeralizatios of Sperer s theorem ad their LYM compaios, ad they geeralize Rota ad Harper s -aalog of Erdős s geeralizatio. Keywords: Sperer s theorem, Meshalki s theorem, LYM ieuality, atichai, r-family, r-chai-free 2000 Mathematics Subject Classificatio. Primary 05D05, 5E20; Secodary 06A07. Ruig head: Projective Meshalki theorem Address for editorial correspodece: Thomas Zaslavsky Departmet of Mathematical Scieces State Uiversity of New York Bighamto, NY U.S.A. To appear i Joural of Combiatorial Theory Series A 2 Research supported by Natioal Sciece Foudatio grat DMS

2 2. Itroducig the Players We preset a theorem that is at oce a -aalog of a geeralizatio, due to Meshalki, of Sperer s famous theorem o atichais of sets ad a geeralizatio of Rota ad Harper s -aalog of both Sperer s theorem ad Erdős s geeralizatio. Sperer s theorem [2 cocers a subset A of P(S), the power set of a -elemet set S, that is a atichai: o member of A cotais aother. It is part (b) of the followig theorem. Part (a), which easily implies (b) (see, e.g., [, Sectio.2) was foud later by Lubell [9, Yamamoto [3, ad Meshalki [0 (ad Bollobás idepedetly proved a geeralizatio [4); coseuetly, it ad similar ieualities are called LYM ieualities. Theorem. Let A be a atichai of subsets of S. The: A ad A A (b) A ( /2 ). (c) Euality occurs i (a) ad (b) if A cosists of all subsets of S of size /2, or all of size /2. The idea of Meshalki s isufficietly well kow geeralizatio 3 (a idea he attributes to Sevast yaov) is to cosider ordered p-tuples A = (A,..., A p ) of pairwise disjoit sets whose uio is S. We call these weak compositios of S ito p parts. Theorem 2. Let M be a family of weak compositios of S ito p parts such that each set M k = {A k : A M} is a atichai. ( ). A M A,..., A p ( ) ( ) (b) M max = α + +α p= α,..., α p p,..., p, p,...,. p (c) Euality occurs i (a) ad (b) if, for each k, M k cosists of all subsets of S of size p, or all of size p. Part (b) is Meshalki s theorem [0; the correspodig LYM ieuality (a) was subseuetly foud by Hochberg ad Hirsch [7. (I expressios like the multiomial coefficiet i (b), sice the lower umbers must sum to, the umber of them that eual p is the least oegative residue of modulo p +.) I [2 Wag ad we geeralized Theorem 2 i a way that simultaeously also geeralizes Erdős s theorem o l-chai-free families: subsets of P(S) that cotai o chai of legth l. (Such families have bee called r-families ad k-families, where r or k is the forbidde legth. We believe a more suggestive ame is eeded.) Theorem 3 ([2, Corollary 4.). Let M be a family of weak compositios of S ito p parts such that each M k, for k < p, is l-chai-free. The: ( ) l p, ad A M A,..., A p 3 We do ot fid it i books o the subject [, 5 but oly i [8.

3 A Meshalki theorem for projective geometries 3 (b) M is o greater tha the sum of the l p largest multiomial coefficiets of the form ( α,...,α p ). Erdős s theorem [6 is essetially the case p = 2, i which A 2 = S \ A is redudat. The upper boud is the the sum of the l largest biomial coefficiets ( j), 0 j, ad is attaied by takig a suitable subclass of P(S). I geeral the bouds i Theorem 3 caot be attaied [2, Sectio 5. Rota ad Harper bega the process of -aalogizig by fidig versios of Sperer s ad Erdős s theorems for fiite projective geometries [. We thik of a projective geometry P = P () of order ad rak (i.e., dimesio ) as a lattice of flats, i which ˆ0 = ad ˆ is the whole set of poits. The rak of a flat a is r(a) = dim a +. The -Gaussia coefficiets (usually the is omitted) are the uatities [! = where! = ( )( ) (). k k! ( k)! They are the -aalogs of the biomial coefficiets. Agai, a family of projective flats is l-chai-free if it cotais o chai of legth l. Let L k be the set of all flats of rak k i P (). Theorem 4 ([, p. 200). Let A be a l-chai-free family of flats i P (). [ l. a A r(a) (b) A is at most the sum of the l largest Gaussia coefficiets [ j for 0 j. (c) There is euality i (a) ad (b) whe A cosists of the l largest classes L k, if l is eve, or the l largest classes ad oe of the two ext largest classes, if l is odd. Our -aalog theorem cocers the projective aalogs of weak compositios of a set. A Meshalki seuece of legth p i P () is a seuece a = (a,..., a p ) of flats whose joi is ˆ ad whose raks sum to. The submodular law implies that, if a J := j J a j for a idex subset J [p = {, 2,..., p}, the a I a J = ˆ0 for ay disjoit I, J [p; so the members of a Meshalki seuece are highly disjoit. To state the result we eed a few more defiitios. If M is a set of Meshalki seueces, the for each k [p we defie M k := {a k : (a,..., a p ) M}. If α,..., α p are oegative itegers whose sum is, we defie the (-)Gaussia multiomial coefficiet to be where α = (α,..., α p ). We write [ [ = = α α,..., α p! α! α p!, s 2 (α) = i<j α i α j for the secod elemetary symmetric fuctio of α. If a is a Meshalki seuece, we write r(a) = (r(a ),..., r(a p ))

4 4 for the seuece of raks. We defie P () to be empty if = 0, a poit if =, ad a lie of + poits if = 2. Theorem 5. Let 0, l, p 2, ad 2. Let M be a family of Meshalki seueces of legth p i P () such that, for each k [p, M k cotais o chai of legth l. The [ a M r(a) s 2 (r(a)) lp, ad (b) M is at most eual to the sum of the l p largest amogst the uatities [ α s 2 (α) for α = (α,..., α p ) with all α k 0 ad α + + α p =. The atichai case (where l = ), the aalog of Meshalki ad Hochberg ad Hirsch s theorems, is captured i Corollary 6. Let M be a family of Meshalki seueces of legth p 2 i P () such that each M k for k < p is a atichai. The [, ad a M r(a) s 2 (r(a)) [ [ (b) M max α α s2(r(a)) = p,..., p, p,..., s2( /p,..., /p, /p,..., /p ). p (c) Euality holds i (a) ad (b) if, for each k, M k cosists of all flats of rak p or all of rak p. We believe but without proof that the largest families M described i (c) are the oly oes. Notice that we do ot place ay coditio i either the theorem or its corollary o M p. Our theorem is ot exactly a geeralizatio of that of Rota ad Harper because a flat i a projective geometry has a variable umber of complemets, depedig o its rak. Still, our result does imply this ad a geeralizatio, as we shall demostrate i Sectio Proof of Theorem 5 The proof of Theorem 5 is adapted from the short proof of Theorem 3 i [3. It is complicated by the multiplicity of complemets of a flat, so we reuire the powerful lemma of Harper, Klai, ad Rota ([8, Lemma 3..3, improvig o [, Lemma o p. 99; for a short proof see [2, Lemmas 3. ad 5.2) ad a cout of the umber of complemets. Lemma 7. Suppose give real umbers m m 2 m N 0, other real umbers,..., N [0,, ad a iteger P with P N. If N k= k P, the () m + + N m N m + + m P. Let m P + ad m P be the first ad last m k s eual to m P. Assumig m P > 0, there is euality i () if ad oly if k = for m k > m P, k = 0 for m k < m P, ad P P = P P.

5 A Meshalki theorem for projective geometries 5 Lemma 8. A flat of rak k i P () has k( k) complemets. Proof. The umber of ways to exted a fixed ordered basis (P,..., P k ) of the flat to a ordered basis (P,..., P ) of P () is k k+ The P k+ P is a complemet ad is geerated by the last k poits i k k k k of the exteded ordered bases. Dividig the former by the latter, there are complemets. (( 2) ( k 2)) ( k 2 ) = k( k) Proof of (a). We proceed by iductio o p. For a flat f, defie ad also, lettig c be aother flat, defie M(f) := {(a 2,..., a p ) : (f, a 2,..., a p ) M} M c (f) := {(a 2,..., a p ) M(f) : a 2 a p = c}. For a M, we write r = r(a ). Fially, C(a ) is the set of complemets of a. If p > 2, the = [ s 2 (r(a)) r r ( r ) s 2 (r(a )) a M [ r(a) a M = a M [ r r ( r ) [ r r ( r ) a M a M(a ). [ r r(a ) c C(a ) a M c (a ) c C(a ) l p 2 [ r r(a ) s 2 (r(a )) by iductio, because M c (a ) is a Meshalki family i c = P r(c) M c k (a ) for k < p, beig a subset of M k+, is l-chai-free, by Lemma 8, = [ r r ( r ) r ( r ) l p 2 a M = P r ad each l l p 2 by the theorem of Rota ad Harper. The iitial case, p = 2, is similar except that the iermost sum i the secod step euals.

6 6 Lemma 9. Let α = (α,..., α p ) with all α k 0 ad α + + α p =. The umber of all Meshalki seueces a i P with r(a) = α is [ α s 2 (α). Proof. If p =, the a = ˆ so the coclusio is obvious. If p >, we get a Meshalki seuece of legth p i P with rak seuece r(a) = α by choosig a to have rak α, the a complemet c of a, ad fially a Meshalki seuece a of legth p i c = P r(c) = P α whose rak seuece is α = (α 2,..., α p ). The first choice ca be made i [ α α ways, the secod i α ( α [ ) ways, ad the third, by iductio, i α α s 2 (α ) ways. Multiply. Proof of (b). Let N(α) be the umber of a M for which r(a) = α. I Lemma 7 take α = N(α) [ ad m s 2 (α) α = α s2(α), [ α ad umber all possible α so that m α m α 2. Lemma 9 shows that all α so Lemma 7 does apply. The coclusio is that N [ [ M = α im α i s 2(α ) + + s 2(α P ), α α P i= where N = ( ) +p p, the umber of seueces α, ad P = mi(l p, N). 3. Strageess of the LYM Ieuality There is somethig odd about the LYM ieuality i Theorem 5(a). A ormal LYM ieuality would be expected to have deomiator [ r(a) without the extra factor s 2 (r(a)). Such a LYM ieuality does exist; it is a corollary of Theorem 5(a); but it is ot strog eough to give the upper boud o M. We prove this weaker ieuality here. Propositio 0. Assume the hypotheses of Theorem 5; that is: 0, l, p 2, ad 2; ad M is a family of Meshalki seueces of legth p i P () such that, for each k [p, M k cotais o chai of legth l. The [ a M r(a) is bouded above by the sum of the l p largest expressios s2(α) for α = (α,..., α p ) with all α k 0 ad α + + α p =. Proof. Agai we apply Lemma 7, this time with α = N(α)/ [ α s 2 (α) ad M α = s 2(α). 4. A Partial Corollary We deduce Theorem 4(a) from the case p = 2 of Theorem 5(a). Our purpose is ot to give a ew proof of Theorem 4 but to show that we have a geeralizatio of it. The key to the proof is that M 2 i our theorem is ot reuired to be l-chai-free. Therefore if we have a l-chai-free set A of flats i P, we ca defie M = {(a, c) : a A ad c C(a)} ; ad M will satisfy the reuiremets of Theorem 5. The LYM sum i Theorem 5(a) the euals the LYM sum i Theorem 4(a), ad we are doe.

7 A Meshalki theorem for projective geometries 7 The same argumet gives a geeral corollary. A partial Meshalki seuece of legth p is a seuece a = (a,..., a p ) of flats i P () such that r(a a p ) = r(a ) + + r(a p ). We simply do ot reuire the joi â = a a p to be ˆ. The geeralized Rota Harper theorem is: Corollary. Let p, l, 2, ad 0. Let M be a family of partial Meshalki seueces of legth p i P () such that, for each k [p, M k cotais o chai of legth l. The [ [ r(â) a M r(â) r(a) s 2 (r(a)) lp ad (b) M is at most eual to the sum of the l p largest amogst the uatities [ α s 2 (α) for α = (α,..., α p+ ) with all α k 0 ad α + + α p+ =. As a special case we geeralize the -aalog of Sperer s theorem. (The -aalog is the case p =.) Corollary 2. Let M be a family of partial Meshalki seueces of legth p i P such that each M k is a atichai. The: [ [ r(â). a M r(â) r(a) s 2 (r(a)) (b) M [ α s 2 (α), i which α = ( p+,..., p+, p+,..., p+ ) where the umber of terms eual to p+ is the least oegative residue of modulo p +. (c) Euality holds i (a) ad (b) if, for each k, M k cosists of all flats of rak p+ or all flats of rak p+. We cojecture that the largest families M described i (c) are uiue. Refereces [ I. Aderso, Combiatorics of Fiite Sets, Claredo Press, Oxford, 987. Corr. repr., Dover, Mieola, N.Y., [2 M. Beck, Xuei Wag, ad T. Zaslavsky, A uifyig geeralizatio of Sperer s theorem, submitted. [3 M. Beck ad T. Zaslavsky, A shorter, simpler, stroger proof of the Meshalki Hochberg Hirsch bouds o compoetwise atichais, J. Combiatorial Theory Ser. A 00 (2002), [4 B. Bollobás, O geeralized graphs. Acta Math. Acad. Sci. Hug. 6 (965), [5 K. Egel, Sperer Theory, Ecyclopedia of Mathematics ad its Applicatios, Vol. 65. Cambridge Uiversity Press, Cambridge, 997. [6 P. Erdős, O a lemma of Littlewood ad Offord, Bull. Amer. Math. Soc. 5 (945), [7 M. Hochberg ad W. M. Hirsch, Sperer families, s-systems, ad a theorem of Meshalki, A. New York Acad. Sci. 75 (970), [8 D. A. Klai ad G.-C. Rota, Itroductio to Geometric Probability, Cambridge Uiversity Press, Cambridge, Eg., 997. [9 D. Lubell, A short proof of Sperer s theorem, J. Combiatorial Theory (966), [0 L. D. Meshalki, Geeralizatio of Sperer s theorem o the umber of subsets of a fiite set (i Russia), Teor. Verojatost. i Primee 8 (963), Eglish tras., Theor. Probability Appl. 8 (963),

8 8 [ G.-C. Rota ad L. H. Harper, Matchig theory, a itroductio, i P. Ney, ed., Advaces i Probability ad Related Topics, Vol., pp Marcel Dekker, New York, 97. [2 E. Sperer, Ei Satz über Utermege eier edliche Mege, Math. Z. 27 (928), [3 K. Yamamoto, Logarithmic order of free distributive lattices, J. Math. Soc. Japa 6 (954),