A unifying generalization of Sperner s theorem

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1 A uifyig geeralizatio of Sperer s theorem Matthias Beck, Xueqi Wag, ad Thomas Zaslavsky State Uiversity of New York at Bighamto matthias@math.bighamto.edu xwag@math.bighamto.edu zaslav@math.bighamto.edu Versio of August 30, 200. Abstract: Sperer s boud o the size of a atichai i the lattice PS of subsets of a fiite set S has bee geeralized i three differet directios: by Erdős to subsets of PS i which chais cotai at most r elemets; by Meshalki to certai classes of compositios of S; by Griggs, Stahl, ad Trotter through replacig the atichais by certai sets of pairs of disjoit elemets of PS. We uify Erdős s, Meshalki s, ad Griggs Stahl Trotter s iequalities with a commo geeralizatio. We similarly uify their accompayig LYM iequalities. Our bouds do ot i geeral appear to be the best possible. Keywords: Sperer s theorem, LYM iequality, atichai, r-family, r-chai-free, compositio of a set Mathematics Subject Classificatio. Primary 05D05; Secodary 06A07. Ruig head: A uifyig Sperer geeralizatio Address for editorial correspodece: Matthias Beck Departmet of Mathematical Scieces State Uiversity of New York Bighamto, NY U.S.A.

2 2. Sperer-type theorems Let S be a fiite set with elemets. I the lattice PS of all subsets of S oe tries to estimate the size of a subset with certai characteristics. The most famous such estimate cocers atichais, that is, subsets of PS i which ay two elemets are icomparable. We let x deote the greatest iteger x ad x the least iteger x. Theorem. Sperer [0]. Suppose A,..., A m S such that A k A j for k j. The m /2. Furthermore, this boud ca be attaied for ay. Sperer s theorem has bee geeralized i may differet directios. Here are three: Erdős exteded Sperer s iequality to subsets of PS i which chais cotai at most r elemets. Meshalki proved a Sperer-like iequality for families of compositios of S ito a fixed umber of parts, i which the sets i each part costitute a atichai. Fially, Griggs, Stahl, ad Trotter exteded Sperer s theorem by replacig the atichais by sets of pairs of disjoit elemets of PS satisfyig a itersectio coditio. I this paper we uify Erdős s, Meshalki s, ad the Griggs Stahl Trotter iequalities i a sigle geeralizatio. However, except i special cases amog which are geeralizatios of the kow bouds, our bouds are ot the best possible. For a precise statemet of Erdős s geeralizatio, call a subset of PS r-chai-free if its chais i.e., liearly ordered subsets cotai o more tha r elemets; that is, o chai has legth r. I particular, a atichai is -chai-free. The geeralizatio of Theorem. to r-chai-free families is Theorem.2 Erdős [3]. Suppose {A,..., A m } PS cotais o chais with r + elemets. The m is bouded by the sum of the r largest biomial coefficiets k, 0 k. The boud is attaiable for every ad r. Note that for r =, we obtai Sperer s theorem. Goig i a differet directio, Sperer s iequality ca be geeralized to certai ordered weak partitios of S. We defie a weak compositio of S ito p parts as a ordered p-tuple A,..., A p of sets A k, possibly void, such that A,..., A p are pairwise disjoit ad A A p = S. A Sperer-like iequality suitable for this settig was proposed by Sevast yaov ad proved by Meshalki see [8]. By a p-multiomial coefficiet for we mea a multiomial coefficiet, where ai 0 ad a + + a p =. Let [p] := {, 2,..., p}. Theorem.3 Meshalki. Let p 2. Suppose A j,..., A jp for j =,..., m are differet weak compositios of S ito p parts such that for all k [p] the set {A jk : j m} igorig repetitio forms a atichai. The m is bouded by the largest p-multiomial coefficiet for. Furthermore, the boud is attaiable for every ad p. The term r-family or k-family, depedig o the ame of the forbidde legth, has bee used i the past, but we thik it is time for a distictive ame.

3 A Uifyig Geeralizatio of Sperer s Theorem 3 This largest multiomial coefficiet ca be writte explicitly as! ρ p +! p ρ, p! where ρ = p p. To see why Meshalki s iequality geeralizes Sperer s Theorem, suppose A,..., A m S form a atichai. The S A,..., S A m also form a atichai. Hece the m weak compositios A j, S A j of S ito two parts satisfy Meshalki s coditios ad Sperer s iequality follows. Yet aother geeralizatio of Sperer s Theorem is Theorem.4 Griggs Stahl Trotter [4] Suppose {A j0,..., A jq } are m differet chais i PS such that A ji A kl for all i ad l ad all j k. The m q/2. Furthermore, this boud ca be attaied for all ad q. A equivalet, simplified form of this result i which A j = A j0, B j = S A jq, ad replaces q is Theorem.4. Let > 0. Suppose A j, B j are m pairs of sets such that A j B j = for all j, A j B k for all j k, ad all A j + B j. The m /2 ad this boud ca be attaied for every. Sperer s iequality follows as the special case i which A,..., A m S form a atichai ad B j = S A j. Theorems.2,.3, ad.4 are icomparable geeralizatios of Sperer s Theorem. We wish to combie ad hece further geeralize these geeralizatios. To state our mai result, we defie a weak partial compositio of S ito p parts as a ordered p-tuple A,..., A p such that A,..., A p are pairwise disjoit sets, possibly void hece the word weak, ad A A p S. If we do ot specify the superset S the we simply talk about a weak set compositio ito p parts this could be a weak compositio of ay set. Our geeralizatio of Sperer s iequality is: Theorem.5. Fix itegers p 2 ad r. Suppose A j,..., A jp for j =,..., m are differet weak set compositios ito p parts with the coditio that, for all k [p] ad all I [m] with I = r +, there exist distict i, j I such that either A ik = A jk or A ik l k A jl A jk l k A il, ad let := max j m A j + + A jp. The m is bouded by the sum of the r p largest p-multiomial coefficiets for itegers less tha or equal to. If r p is larger tha r+p p, the umber of p-multiomial coefficiets, the we regard the sequece of coefficiets as exteded by 0 s.

4 4 We heartily agree with those readers who fid the statemet of this theorem somewhat ureadable. We would first like to show that it does geeralize Theorems.2,.3, ad.4 simultaeously. The last follows easily as the case r =, p = 2. Theorem.3 ca be deduced by choosig r = ad restrictig the weak compositios to be compositios of a fixed set S with elemets. Fially, Theorem.2 follows by choosig p = 2 ad the weak compositios to be compositios of a fixed -set ito 2 parts. What we fid more iterestig, however, is that specializatios of Theorem.5 yield simply stated corollaries that combie two at a time of Theorems.2,.3, ad.4. Sectio 4 collects these corollaries. The coditio of the theorem implies that each set A k = {A jk : j [m]} igorig repetitio is r-chai-free. We suspect that the coverse is ot true i geeral. It is true if all the weak set compositios are weak compositios of the same set of order, as i Corollary 4.. All the theorems we have stated have each a slightly stroger compaio, a LYM iequality. I Sectio 2, we state these iequalities ad show how Theorems..5 ca be deduced from them. The proofs of Theorem.5 ad the correspodig LYM iequality are i Sectio 3. After the corollaries of Sectio 4, i Sectio 5 we show that some, at least, of our upper bouds caot be attaied. 2. LYM iequalities I attemptig to fid a ew proof of Theorem., Lubell, Yamamoto, ad Meshalki idepedetly came up with the followig refiemet: Theorem 2. Lubell [7], Yamamoto [], Meshalki [8]. Suppose A,..., A m S such that A k A j for k j. The. A k Sperer s iequality follows immediately by otig that max k k k= = /2. A LYM iequality correspodig to Theorem.2 appeared to our kowledge first i [9]: Theorem 2.2 Rota Harper. Suppose {A,..., A m } PS cotais o chais with r + elemets. The r. A k k= Deducig Erdős s Theorem.2 from this iequality is ot as straightforward as the coectio betwee Theorems 2. ad.. It ca be doe through Lemma 3., which we also eed i order to deduce Theorem.5.

5 A Uifyig Geeralizatio of Sperer s Theorem 5 The LYM compaio of Theorem.3 first appeared i [5]; agai, Meshalki s Theorem.3 follows immediately. Theorem 2.3 Hochberg Hirsch. Suppose A j,..., A jp for j =,..., m are differet weak compositios of S ito p parts such that for each k [p] the set {A jk : j m} igorig repetitios forms a atichai. The. A j,..., A jp The LYM iequality correspodig to Theorem.4 is due to Bollobás. Theorem 2.4 Bollobás [2]. Suppose A j, B j are m pairs of sets such that A j B j = for all j ad A j B k for all j k. The Aj + B j A j. Oce more, the correspodig upper boud, the Griggs Stahl Trotter Theorem.4, is a immediate cosequece. Naturally, there is a LYM iequality accompayig our mai Theorem.5. Like its sibligs, it costitutes a refiemet. Theorem 2.5. Let p 2 ad r. Suppose A j,..., A jp for j =,..., m are differet weak compositios of ay sets ito p parts satisfyig the same coditio as i Theorem.5. The Aj + + A jp A j,..., A jp r p. Example 2.. The complicated hypothesis of Theorem 2.5 caot be replaced by the assumptio that each A k is r-chai-free, because the there is o LYM boud idepedet of. Let p 2, S = [], ad A = {A, {}, { },..., { p + 2} : A A } where A is a largest r-chai-free family i [ p + ], specifically, A = j I P j [ p + ] where { p + r p + r p + r I =, +, } + r.

6 6 The LYM sum is A +p A A A,,..., There is o possible upper boud i terms of. = A! A + p! A A = p + j! j j + p! j I = p + p j + 2 p + j! j I as. 3. Proof of the mai theorems Proof of Theorem 2.5. Let S be a fiite set cotaiig all A jk for j =,..., m ad k =,..., p, ad let = S. We cout maximal chais i PS. Let us say a maximal chai separates the weak compositio A,..., A p if there exist elemets = X 0 X l X lp = S of the maximal chai such that A k X lk X lk for each k. There are 2 A + + A p A! A p! A A p! maximal chais separatig A,..., A p. To prove this, replace maximal chais {x } {x, x 2 } S by permutatios x, x 2,..., x of S. Choose A + + A p places for A A p ; the arrage A i ay order i the first A of these places, A 2 i the ext A 2, etc. Fially, arrage S A A p i the remaiig places. This costructs all maximal chais that separate A,..., A p. We claim that every maximal chai separates at most r p weak partial compositios of S. To prove this, assume that there is a maximal chai that separates N weak partial compositios A j,..., A jp. Cosider all first compoets A j ad suppose r + of them are differet, say A, A 2,..., A r+,. By the hypotheses of the theorem, there are i, i [r + ] such that A i meets some A i l where l > ad A i meets some A il where l >. By separatio, there are q ad q such that A i X q X 0 ad A i X q X 0, ad there are q l, q l, q l, q l such that q q l q l, q q l q l, ad A il X ql X ql ad A i l X q l X q l. Sice A i meets A i l, there is a elemet a i X q l X q l ; it follows that q l < q. Similarly, q l < q. But this is a cotradictio. It follows that, amogst the N sets A j, there are at most r differet sets. Hece by the pigeohole priciple there are N/r amog the N weak partial compositios that have the same first set A j.

7 A Uifyig Geeralizatio of Sperer s Theorem 7 Lookig ow at these N/r weak partial compositios, we ca repeat the argumet to coclude that there are N/r /r N/r 2 weak partial compositios for which both the A j s ad the A j2 s are idetical. Repeatig this process p times yields N/r p weak partial compositios ito p parts whose first p parts are idetical. But ow the hypotheses imply that the last parts of all these weak partial compositios are at most r differet sets; i other words, there are at most r distict weak partial compositios. Hece N/r p r, whece N r p. If we kow that all the compositios are weak but ot partial compositios of S, the the last parts of all these N/r p weak compositios are idetical. Thus N r p. Sice at most r p weak partial compositios of S are separated by each of the! maximal chais, from 2 we deduce that r p! A j! A jp! A j A jp! A j + + A jp The theorem follows. =! Aj + + A jp A j,..., A jp To deduce Theorem.5 from Theorem 2.5, we use the followig lemma, which origially appeared i somewhat differet ad icomplete form i [9], used there to prove Erdős s Theorem.2 by meas of Theorem 2.2, ad appeared i complete form i [6, Lemma 3..3]. We give a very short proof, which seems to be ew. Lemma 3. Harper Klai Rota. Suppose M,..., M N R satisfy M M 2 M N 0, ad let R be a iteger with R N. If q,..., q N [0, ] have sum the q + + q N R, q M + + q N M N M + + M R.. Proof. By assumptio, N q k k=r+ k= R q k. Hece, by the coditio o the M k, N N R q k M k M R q k M R q k k=r+ k=r+ k= which is equivalet to the coclusio. R q k M k, k= Proof of Theorem.5. Let S be ay fiite set that cotais all A jk. Write dow the LYM iequality from Theorem 2.5.

8 8 From the m weak partial compositios A j,..., A jp of S, collect those whose shape is a,..., a p ito the set Ca,..., a p. Label the p-multiomial coefficiets for itegers as M, M 2,... so that M M 2. If M k is, let q k := Ca,..., a p /M k. By Theorem 2.5, the q k s ad M k s satisfy all the coditios of Lemma 3. with N replaced by the umber of p-tuples a,..., a p whose sum is at most, that is +p p, ad R replaced by min, r p. Hece Ca,..., a p M + + M R. a + +a p The coclusio of the theorem ow follows, sice m = Ca,..., a p. a + +a p 4. Cosequeces As promised i Sectio, we ow state special cases of Theorems.5/2.5 that uify pairs of Theorems.2,.3, ad.4 as well as their LYM compaios. The first special case uifies Theorems.2/2.2 ad.3/2.3. It is a corollary of the proof of the mai theorems, ot of the theorems themselves. See [] for a short, direct proof. Corollary 4.. Suppose A j,..., A jp are m differet weak compositios of S ito p parts such that for each k [p ], the set {A jk : j m} is r-chai-free. The r p. A j,..., A jp Cosequetly, m is bouded by the sum of the r p largest p-multiomial coefficiets for. Proof. We ote that, for a family of m weak compositios of S, the coditio of Theorem 2.5 for a particular k [p ] is equivalet to {A jk } j beig r-chai-free. Thus by the hypothesis of the corollary, the hypothesis of the theorem is met for k =,..., p. The the proof of Theorem 2.5 goes through perfectly with the oly differece, explaied i the proof, that eve without a coditio o k = p we obtai N r p. I the proof of Theorem.5, uder our hypotheses the sets Ca,..., a p with a + + a p < are empty. Therefore we take oly the p-multiomial coefficiets for, labelled M M 2. I applyig Lemma 3. we take R = min, r p ad summatios over a + + a p =. With these alteratios the proof fits Corollary 4.. A good way to thik of Corollary 4. is as a theorem about partial weak compositios, obtaied by droppig the last part from each of the weak compositios i the corollary.

9 A Uifyig Geeralizatio of Sperer s Theorem 9 Corollary 4.2. Fix p 2 ad r. Suppose A j,..., A jp are m differet weak partial compositios of a -set S ito p parts such that for each k [p], the set {A jk : j m} is r-chai-free. The m is bouded by the sum of the r p largest p + -multiomial coefficiets for. A differece betwee this ad Theorem.5 is that Corollary 4.2 has a weaker ad simpler hypothesis but a much weaker boud. But the biggest differece is the omissio of a accompayig LYM iequality. Corollary 4. obviously implies oe, but it is weaker tha that i Theorem 2.5 because, sice the top umber i the latter ca be less tha, the deomiators are much smaller. We do ot preset i Corollary 4.2 a LYM iequality of the kid i Theorem 2.5 for the very good reaso that oe is possible; that is the meaig of Example 2.. The secod specializatio costitutes a weak commo refiemet of Theorems.2/2.2 ad.4/2.4. We call it weak because its specializatio to the case B j = S A j, which is the situatio of Theorems.2/2.2, is weaker tha those theorems. Corollary 4.3. Let r be a positive iteger. Suppose A j, B j are m pairs of sets such that A j B j = ad, for all I [m] with I = r +, there exist distict i, j I for which A j B k A k B j. Let = max j A j + B j. The Aj + B j A j r. Cosequetly, m is bouded by the sum of the r largest biomial coefficiets k for 0 k. This boud ca be attaied for all ad r. Proof. Set p = 2 i Theorems.5/2.5. To attai the boud, let A j rage over all k-subsets of [] ad let B j = [] A j. The last special case of Theorems.5/2.5 we would like to metio is that i which r = ; it uifies Theorems.3/2.3 ad.4/2.4. Corollary 4.4. Suppose A j,..., A jp are m differet weak set compositios ito p parts with the coditio that, for all k [p] ad all distict i, j [m], either A ik = A jk or A ik l k A jl A jk l k A il. ad let max j Aj + + A jp. The Aj + + A jp A j,..., A jp. Cosequetly, m is bouded by the largest p-multiomial coefficiet for. The boud ca be attaied for every ad p.

10 0 Proof. Everythig follows from Theorems.5/2.5 except the attaiability of the upper boud, which is a cosequece of Theorem The maximum umber of compositios Although the bouds i all the previously kow Sperer geeralizatios of Sectio ca be attaied, for the most part that seems ot to be the case i Theorem.5. The key difficulty appears i the combiatio of r-families with compositios as i Corollary 4.. We thik it makes o differece if we allow partial compositios but we have ot proved it. We begi with a refiemet of Lemma 3.. A weak set compositio has shape a,..., a p if A k = a k for all k. Lemma 5.. Give values of, r, ad p such that r p +p p, the boud i Corollary 4. ca be attaied oly by takig all weak compositios of shape a,..., a p that give p- multiomial coefficiet larger tha the r p +-st largest such coefficiet M r p +, ad oe whose shape gives a smaller coefficiet tha the r p -st largest such coefficiet M r p. Proof. First we eed to characterized sharpess i Lemma 3.. Our lemma is a slight improvemet o [6, Lemma 3..3]. Lemma 5.2. I Lemma 3., suppose that M R > 0. The there is equality i the coclusio if ad oly if q k = if M k > M R ad q k = 0 if M k < M R ad also, lettig M R + ad M R be the first ad last M k s equal to M R, q R q R = R R. I Lemma 5., all M k > 0 for k +p p. We assume N is o larger tha +p p. The cotrary case is easily derived from that oe. It is clear that, whe applyig Lemma 3., we have to have i our set of weak compositios all those of the shapes a,..., a p for which > Mr p ad oe for which < Mr p. The rest of the m weak compositios ca have ay shapes for which = Mr p. If M r p > M r p + this meas we must have all weak compositios with shapes for which > Mr p +. To explai why the boud caot usually be attaied, we eed to defie the first appearace of a size a i i the descedig order of p-multiomial coefficiets for. Fix p 3 ad ad let = νp + ρ where 0 ρ < p. I, the ai are the sizes. The multiset of sizes is the form of the coefficiet. Arrage the multiomial coefficiets i decreasig order: M M 2 M 3. There are may such orderigs; choose oe arbitrarily, fix it, ad call it the descedig order of coefficiets. Thus, for example, M = > M 2 = = M 3 = = M pp + if p ν,..., ν ν +, ν,..., ν, ν

11 A Uifyig Geeralizatio of Sperer s Theorem sice M 3,..., M pp + have the same form as M 2, ad M = ν +,..., ν = = M p ρ > M if p, p ρ+ where the form of M has ρ sizes equal to ν +, so M,..., M p all have the same form. ρ As we sca the descedig order of multiomial coefficiets, each possible size κ, 0 κ, appears first i a certai M i. We call M i the first appearace of κ ad label it L κ. For example, if p, L ν = M > L ν+ = L ν = M 2, while if p the L ν = L ν+ = M. It is clear that L ν > L ν >... ad L ν+ > L ν+2 >..., but the way i which the lower L κ s, where κ ν, iterleave the upper oes is ot obvious. We write L k for the k-th L κ i the descedig order of multiomial coefficiets. Thus L = L ν ; L 2 = L ν+ ad L 3 = L ν or vice versa if p, ad L 2 = L ν+ if p while L 3 = L ν+2 or L ν. Theorem 5.3. Give r 2, p 3, ad p, the boud i Corollary 4. caot be attaied if L r > M r p +. The proof depeds o the followig lemma. Lemma 5.4. Let r 2 ad p 3, ad let κ,..., κ r be the first r sizes that appear i the descedig order of p-multiomial coefficiets for. The umber of all coefficiets with sizes draw from κ,..., κ r is less tha r p ad their sum is less tha M + + M r p. Proof. Clearly, κ,..., κ r form a cosecutive set that icludes ν. Let κ be the smallest ad κ the largest. Oe ca verify that, i κ,...,κ,x ad κ,...,κ,y, it is impossible for both x ad y to lie i the iterval [κ, κ ] as log as r p 2 > 0. Proof of Theorem 5.3. Suppose the upper boud of Corollary 4. is attaied by a certai set of weak compositios of S, a -elemet set. For each of the first r sizes κ,..., κ r that appear i the descedig order of p-multiomial coefficiets, L κi has sizes draw from κ,..., κ r ad at least oe size κ i. Takig all coefficiets M k that have the same forms as the L κi, κ i will appear i each positio j i some M k. By hypothesis ad Lemma 5., amog our set of weak compositios, every κ i -subset of S appears i every positio i the weak compositios. If ay subset of S of a differet size from κ,..., κ r appeared i ay positio, there would be a chai of legth r i that positio. Therefore we ca oly have weak compositios whose sizes are amog the first r sizes. By Lemma 5.4, there are ot eough of these to attai the upper boud. Theorem 5.3 ca be hard to apply because we do ot kow M r p +. O the other had, we do kow L κ sice it equals κ,a 2,...,a p where a2,..., a p are as early equal as possible. A more practical criterio for oattaimet of the upper boud is therefore Corollary 5.5. Give r 2, p 3, ad p, the boud i Corollary 4. caot be attaied if L r > L r+.

12 2 Proof. It follows from Lemma 5.4 that L r+ is oe of the first r p coefficiets. Thus L r > L r+ M r p + ad Theorem 5.3 applies. It seems clear that L r will almost always be larger tha L r+ if r 3 or p so our boud will ot be attaied. However, cases of equality do exist. For istace, take p = 3, r = 3, ad = 0; the L 5 = L = 0 5,4, = 260 ad L 6 = L 6 = 0 6,2,2 = 260. Thus if r = 5, Corollary 5.5 does ot apply here. We thik the boud is still ot attaied but we caot prove it. We ca isolate the istaces of equality for each r, but as r grows larger the calculatios quickly become extesive. Thus we state the results oly for small values of r. Propositio 5.6. The boud i Corollary 4. caot be attaied if 2 r 5 ad p 3 ad r, except possibly whe r = 2, p, ad p = 3, 4, 5, or whe r = 4, p 4, ad = 2p, or whe r = 5, p = 3, ad = 0. Proof sketch. Suppose p. We have verified by log but routie calculatios which we omit that L = L 2 > L 3 > L 4 > L 5 > L 6 except that L 4 = L 5 if ρ = p ad p 4 ad ν = ad L 5 = L 6 whe p = ν = 3 ad ρ =. If p the L > L 2 = L 3 > L 4 > L 5 > L 6. This implies the propositio for r = 3, 4, or 5. We approach r = 2 differetly. The largest coefficiets are M = > M 2 = = = M pp + > M pp +2. ν,..., ν ν +, ν,..., ν, ν If pp + r p, the boud is uattaiable by Theorem 5.3. That is the case whe p 6. Refereces [] M. Beck ad T. Zaslavsky, A short geeralizatio of the Meshalki Hochberg Hirsch bouds o compoetwise atichais. I preparatio. [2] B. Bollobás, O geeralized graphs. Acta Math. Acad. Sci. Hug , [3] P. Erdős, O a lemma of Littlewood ad Offord. Bull. Amer. Math. Soc , [4] J. R. Griggs, J. Stahl, ad W. T. Trotter, A Sperer theorem o urelated chais of subsets. J. Combiatorial Theory Ser. A , [5] M. Hochberg ad W. M. Hirsch, Sperer families, s-systems, ad a theorem of Meshalki. A. New York Acad. Sci , [6] D. A. Klai ad G.-C. Rota, Itroductio to Geometric Probability. Cambridge Uiversity Press, Cambridge, Eg., 997. [7] D. A. Lubell, A short proof of Sperer s theorem. J. Combiatorial Theory 966, [8] 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 , [9] 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. [0] E. Sperer, Ei Satz über Utermege eier edliche Mege. Math. Z , [] K. Yamamoto, Logarithmic order of free distributive lattices. J. Math. Soc. Japa 6 954,