*m[ mmm Sir WHflVil rffm MB HBmiHn BjH SB PUHfl H UHSa BHMnHHnBff ■BBflBBHMHSMMB! P HH Smil H MHW iUMlag H Wj; ■ HB ■■1 yv wUMKBHi mtSm US — v A. Huh RmS — UMUil BE I unyHSHM gui una nRHaHH BraM§fflHi ■BHIC&MJEin RSfl 0] IMi is^ WmBtimm LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 5"! 0.84 IQ6r ho. £49-654 cop. 2 The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN SEP 2 7 'ECO L161 — O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/spernerstheoremo650liuc 3/0. t . . . zd a. is upperbounded Vl X l X 2 Vl by the sum of the £ largest binomial coefficients of order n. Kleitman [3] and Katona [k] improved Sperner 's result in the following way: Let S-, US = S and S f) S = be a partition of S. Let F = (A,, A ,. . . , A., ...} be a family of subsets of S such that no two subsets A. and A. 1 1 j in F Q possess the properties: < (i) (A. n S 1 ) = (A n S ± ) and (A. fl Sg) => (A. fl Sg); or (ii) (A. fl S 1 ) => (A n S 1 ) and (A. fl Sg) = (A fl Sg) Then DeBruijn, Tengbergen, and Kruyswijk [k~\ generalized Sperner's result in another direction: Let N be a positive integer and let p, p . . . p be a prime factor decomposition of N. We define the degree of an integer to be the sum of the exponents of the prime [kj denotes the largest integer not larger than k. -2- f actors of the integer. We use the notation S (N) to denote the m number of divisors of N that are of degree i, where m is the degree of N. We follow the convention that S (n) = 1 for i = and m S (N) = for i < and i > m. Let F. = {a.. ,a_,. . . ,a. ,. . . } be a m v 1' 2.' ' i' set of divisors of N such that no two divisors a. and a. in F i 3 possess the property a. |a.. According to DeBruijn, Tengbergen, and Kruyswijk, |F | « S {^ 2 J(N) Recently, the result of DeBruijn, Tengbergen, and Kruyswijk was generalized by Schonheim [6]. Katona [7] obtained a general result that includes all the results mentioned above as special cases. In this paper, we present a result that is similar to that of Katona' s. Our result is more general in that it is applicable to the direct products of arbitrary partially ordered sets, yet Katona' s result is restricted to direct products of "symmetrical chain graphs". For the lattice of subsets of a set and the lattice of divisors of an integer, we obtain simple proofs of many of the known results, and also are able to sharpen some of them. Our main result is Theorem 3. However, for exposition purpose we show first some special cases. 2. Product of partially ordered sets Let P as {p ,p ,. . . ,p. ,. .. ) be an arbitrary partially ordered set. Let L, denote the set of integers {0,1,2,. .. ,h) ordered by the "larger than or equal to" relation. Let R = P x L. The elements in R shall be denoted (p.,j), where p. e P and j e (0,1,2,. .. ,h}. Let -3- T c R. An element (p.,j) in T is said to be an unobstructed element in T if there is no other element (p. l ,j') in T such that p. > p! and j = y or p.. = p.' and j > j 1 . We also introduce the notation, for a fixed j, • T(J) = {(p^d) I (p^J) e T} T(j) = ib ± ,y) | is ± ,y) eT, j'>j) t(I) = {(p^j) | (p., j') e t, y > y We shall use P[£] to denote a maximal-sized subset of P such that no 1+1 elements in the subset form a chain of length £+1. It will be understood that P[iJ = f for £ < 0. Theorem 1 ; Let R = P x L. Let F_ be a subset of R such that no £+1 elements in F n form a chain of length £+1. Then for i > h h |F | ^ Z |P[£+h-2t]| f or £ g h t=0 £-1 |F n | * Z |P[£+h-2t] U t=0 Proof : We prove the case £ > h. The case £ ^ h can be proved in a similar manner. Let M n denote the set of all unobstructed elements in F . We note that (p.. ,j) e M (o) implies that (p.,j') / M (0) f° r y £ 3' Therefore, there is a one-to-one correspondence between the elements in M^O) and M (o). That is, 1^(0)1 = |m (o)| -1^ We also note that (p.,o) e Mq(o) implies that (p.,0) L(0) does not contain chains of length larger than h, we have |f (o) U ^(0)| g |P[f+h]| Similarly, for t = l,2,...,h-l, since the set F, (t) does not contain chains of length larger than !-t and the set M, (t) does not contain Because every element of P, (t) dominates a chain of length t. -5- chains of length larger than h-t , we have |F t (t) U M t (t)| £ |P[l+h-2t]| Finally, because F, does not contain chains of length larger than £-h, we have , |F h | g |P[£-h]| We thus obtain h |FJ £ Z |P[£+h-2t]| I t=0 Corollary 1.1 : Let R = P x L, . Let F_ be a subset of R such that no £+1 elements in F form a chain of length i+1. Then |F Q | * |P[l+l]| + |P[!-1]| From Corollary 1.1 we obtain Erdos ' extension of Sperner's result: Corollary 1. 2 : In the lattice of subsets of a finite set of n elements, the size of a family of subsets that does not contain a chain of length i+1 is upperbounded by the sum of the £ largest binomial coefficients of order n. Proof : Let R = (L. ) x K. . We can then prove the corollary by induction on n, using the fact that the sum of the £+1 largest binomial coefficients of order n-1 and the £-1 largest binomial If (p.;*.]) and (p.',j) are two elements in M (t), then they must be J- x u incomparable. Consequently, the corresponding elements (p.,t) and (p.* f t) in M, (t) must also be incomparable. X "0 -6- coefficients of order n-1 is equal to the sum of the I largest binomial coefficients of order n. ■ Similarly, we can obtain Schbnheim's extension of DeBruijn, Tengbergen, and Kruyswijk's result. Corollary 1. 5 : In the lattice of divisors of an integer N, the size of a family of divisors that does not contain a chain of length l+l is upperbounded by the sum of the i largest values of S (N), where m is the degree of N. °a a 2 a n-l a n Proof: Let N = p. p„ c . . . p 7 p . Let R = (L x L x *± *2 n-1 *n 0^ a . . . X L ) x L . We can prove the corollary by induction on n, using n-J. n i a the result that the sum of the I largest values of S + ^(Np ) is equal to a Z [sum of the £+a-2t values of S 1 (N)] « m t=0 for I > a, and is equal to i-1 i Z [sum of the £+a-2t values of S (N) ] for I % a. This result can be obtained routinely from the facts: (i) S (N), S (n), ..., S <■ ' ■" (N) is a non-decreasing sequence (ii) S J (N) = S m - j (N) m m (iii) S^ (Np°) = S J (N) + S^CN) + S J *' 2 (N) + ... + S J '" a (N) m+a m m m m Examining the proof of Theorem 1, we realize that the condition stated in Theorem 1 can be weakened: T This result comes directly from the relation: • - a + m -7- Theorem 2 : Let R = P X L. Let F be a subset of R such that there are no £+1 elements (p i , j ± ) , (p ± , j 2 ), . .., (p jL , j^), (p. , j , ) in F A possessing the properties: for some t, £ t £ h, £+1 (i) p. > p. > ... > P. and X l X 2 X i-t+l J x = 2 = ••• = J|_t+1 = t; (ii) for k = £-t+l, ..., £, either p > p. and x k k+1 j k = J k+1 ' k k+1 Then for £ > h for £ ^ h h |F | £ Z |P[£+h-2t] u t=0 £-1 |F n | £ E |P[£+h-2t]| t=0 Corollary 2.1 : Let F Q = {A,, A ,. .. ,A. ,. . . ) be a family of subsets in the lattice of subsets of a finite set S of n elements. Suppose that there are no £+1 subsets A. , A. , ..., A. in F X l X 2 £+1 such that either A. zd A. o ... 3 A. and they all contain a X l X 2 X £ + l certain element a, a e S: or A. 3 A. 3 ... z> A. and they ail do X l 2 £+1 not contain the element a: or A. =3 A. 3 ... 3 A. and they all contain i i 1 2 £ the element a and A. = A. - {a}. Then the size of F n is upperbounded X £+l X £ ° by the sum of the £ largest binomial coefficients of order n. -8- Corollary 2.1 sharpens Frdos' result. If we set I = 1 in Corollary 2.1 we obtain a special case of Kleitman and Katona's result. Specifically, this case corresponds to partitioning S into S. and S such that S.. = {a} and S = S - (a). We shall present a more general result in Corollary 3»1» Similarly, Schonheim's result as stated in Corollary 1. 5 can "be sharpened: Corollary 2.2 : Let F = {a,, a ,.. . ,a. ,. . .) be a collection of integers in the lattices of divisors of an integer N = p., p ... p -i-2 n such that there are no i+1 divisors a. , a. , ..., a. in F possessing X l X 2 X i+1 the properties: for some t, ^ t ^ a , (i) a, 4- a 4- ... -s- a , + , and they all contain exactly the t^h power of p for ^ t ^ a ; * n n' (ii) for k k |-t+l, ..., l f a. 4- a. and the quotient x k x k+l of a. divided by a. either does not contain a \ X k+1 power of p or is only a power of p . Then the size of F is upperbounded by the sum of the £ largest values of S (N) where m is the degree of N. Let P and Q be two partially ordered sets. Let R = P x Q. Suppose that the elements in Q are partitioned into d disjoint chains. We write a. 4- a. to mean a. is divisible by a.. 1 J 1 J -9- Let h, , h ..... h, denote the lengths of these chains. A direct 1' 2 d consequence of Theorem 2 is Theorem 3 ' Let R be a subset of R such that no 1+1 elements in F n , (p. , q. ), (P, , q i ), ..., (p.. , q i ) possess the properties: x l J l x 2 J 2 1+1 J i+1 (i) (P, ,q, ) > (P, ,q, )>...> (p, ,q. )j x l J l 2 J 2 1+1 J £+l (ii) q. , q. , . .., q- are in the same chain in the °1 J 2 J l+1 partition of Q into disjoint chains; th Then (iii) if in the chain containing q. , q. is the t °1 J l element, q. = q. = ... = q. ; J l J 2 J £-t+l (iv) for k = 1-t+l, . .., I 3 either p. > p. and \ X k+1 q. =q. , or p. =p. and q . > q . . J k J k+1 x k x k+l J k J k+1 d ■ | * Z z. o 1 i=l 1 where for 1 > h. for f gh. i h. l Z. = Z |P[f4h.-2t]| t=0 1-1 Z = Z |P[!+h.-2t]| t=0 x In order to apply Theorems 3 to the lattice of subsets of a finite set, we define a canonical partition of (L ) n into disjoint The elements in a chain are labelled as the th , 1st, 2 nd , ..., t th , ... elements, starting at the bottom of the chain. -10- chains recursively as follows: (i) 1^ is partitioned into a chain of length 2. (ii) Let p. > p. > ... > p. be a chain in a canonical 1 2 m partition of (L.. ) . Then ,0) (p ± A) > (p, ,o) > ( P ,o) > ... > ( P ,o) > ( P . X l x l x 2 Vl 1 m (P ± A) >(P. ,1) >... > (P, A) 2 3 m will be two chains in a canonical partition of (L.) . k n Let P m (L ) , Q ■= (L ) , and R m P x Q, By induction on n, we can immediately show that corresponding to a canonical partition of Q, into d disjoint chains, the sum E Z. in Theorem 3 is equal to the sum of the i=l x largest I binomial coefficients of order k+n. We thus obtain: Corollary 3.1 : Let S be a finite set of size n. Suppose S is partitioned into S 1 and S 2 such that S, Us. = S and S f) S p = 0. Let us partition the subsets of Sp into disjoint chains according to a canonical partition. Let F be a family of subsets of S such that there are no |+1 subsets A. , A. , . . . , A. in F possessing the X l X 2 Vl properties: (i) A. 3 A. 3 ... r> A. X l X 2 Vl (ii) A. OS., A. n S , ..., A. PI S are in X l 2 d Vl d the same chain in the canonical partition of Q, (iii) if in the chain containing A. DS^, A. D S„ is & i 1 2' i 1 2 the t th element, A. (1 S = A. D S„ = ... = A. D S n X l 2 X 2 2 Vt+1 2 t These indeed are what are known as symmetric ch ains defined by DeBruiin Tengbergen, and Kruyswijk. -11- (iv) for k «= 1-t+l, ..., I, either A. S. => A. fl S n and A. S = A. D S„, or \ 1 X k + 1 X \ 2 W 2 ' A. n S. = A. D S n and A. fl S 3 A. OS. \ 1 X k + 1 X x k 2 W 2 Then the size of F is upperbounded by the sum of the I largest binomial coefficients of order n. For 1=1, Corollary 3.1 is reduced to: Let F Q be a family of subsets of S such that no two subsets A. and A. in F_ possess the i J properties (i) A. fl S„ and A. fl S„ are in the same chain in a v ' l 2 J 2 canonical partition of the lattice of subsets of ^ S ; and (ii) either (A. HSJd (A . f) S . ) and (A. D S) «= (A. D S), or v ' l 1 j 1 l 2 j 2 (A ± fl S^ m (Aj fl S 1 ) and (A ± D Sg) 3 (A^ n S 2 ). Then the size of F is upperbounded by the largest binomial coefficient of order n. Clearly, Theorem 3 also leads to a generalization of DeBruijn, Tengbergen, and Kruyswijk's result in the same sense as Corollary 3*1 generalizes Sperner's result. We shall leave the details to the reader. 3 . Remarks It is interesting to note that our approach is quite similar to Kleitman's approach [8] in solving Littlewood and Offord's problem on the distributions of linear combinations of vectors. The author is indebted to D. J. Kleitman and C. Greene for their suggestions and comments. -12- References 1. E. Sperner, Ein Satz uber Untermengen einer en&lichen Menge, Math. Z^ 27 (1928), 5M+-548. 2. P. Erdos, On a lemma of Littlewood and Of ford, Bull. Amer. Math. Soc. 51 (1945), 898-902. 3. D. J, Kleitman, On a lemma of Littlewood and Offord on the . distribution of certain sums, Math. Z. 90 (1965), 251-259. 1*. G. 0. H. Katona, On a conjecture of Erdos and a stronger form of Sperner's theorem, Studia Sci. Math. Hungar. 1 (1966), 59-63. 5. N. G. DeBruijn, C. A. van E. Tengbergen, and D. Kruyswijk, On the set of divisors of a number, Nieuw Arch. Wisk. (2), 23 (19^9-51) } 191-193. 6. J. Schonheim, A generalization of results of P. Erdos, G. Katona and D. Kleitman concerning Sperner's theorem, J. Combinatorial Theory 11 (1971), 111-117. 7. G. 0. H. Katona, A generalization of some generalizations of Sperner's theorem, J. Combinatorial Theory (B) 12 (1972), 72-81. 8. D. J. Kleitman, On a lemma of Littlewood and Offord on the distribution of linear combinations of vectors, Advances in Math. 5 (197°), 155-157 • BIBLIOGRAPHIC DATA SHEET 1. Report No. UIUCDCS-R-T4-650 4. Title and Subtitle SPERNER'S THEOREM ON MAXIMAL-SIZED ANTICHAINS AND ITS GENERALIZATION 3. Recipient's Accession No. 5. Report Date June 1971+ 7. Author(s) C. L. Liu 8. Performing Organization Rept. No. 9. Performing Organization Name and Address Department of Computer Science University of Illinois Urbana, Illinois 6l801 10. Project/Task/Work Unit No. 11. Contract/Grant No. GJ-i)-1538 12. Sponsoring Organization Name and Address National Science Foundation Washington, D . C . 13. Type of Report & Period Covered 14. 15. Supplementary Notes 16. Abstracts This paper generalizes previous work by Sperner [1], Erdbs [2], Kleitman [3], Katona [k], DeBruijn, Tengbergen, and Kruyswijk [5], and Schbnheim [6]. Such generalization not only leads to simple proofs of known results but also sharpens same of these results. 17. Key Words and Document Analysis. 17o. Descriptors 17b. Identifiers /Open-Ended Terms 17e. COSATI Field/Group 18. Availability Statement 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 22. Price FORM NTIS-35 ( 10-70) USCOMM-DC 40329-P7I ** *>, , . ■