Erdös, P.L. (Péter)

Seress, A.

Székely, L.A.

English

CWI's Institutional Repository

On Intersecting Chains in Boolean Algebras PETER L. ERDOS\ AKOS SEREss:t: and LASZLO A. SZEKEL y§ tcentrum voor Wiskunde en Informatica, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands +The Ohio State University, Columbus, OH 43210 §university of New Mexico, Albuquerque, NM 87131 Analogues of the Erdos-Ko-Rado theorem are proved for the Boolean algebra of all subsets of { 1, ... n) and in this algebra truncated by the removal of the empty set and the whole set. 1. Introduction One of the basic results in extremal set theory is the Erdos-Ko-Rado (EKR) Theorem [5]: if :ff' is an intersecting family of k-element subsets of [l, n] = { 1, 2, ... , n} (i.e. every two members of ff have non-empty intersection) and n ~ 2k, then lffl ,::::; G:::D and this bound is attained. We can consider k-subsets of [1, n] as length-k chains in the (total) order 1 < 2 < ... < n: using this terminology, the EKR theorem is a result about intersecting k-chains in a special partially ordered set. Erdos, Faigle, and Kern [3] pointed out that certain results of Deza, Frankl [2, Theorem 5.8], and Frankl and Fliredi [7] on intersecting sequences of integers may be interpreted as results on intersecting families of chains in some partially ordered sets. The purpose of this note is to prove analogues of the EKR theorem in two other partially ordered sets: in the Boolean algebra &4n of all subsets of [1,n] (with A ,::::; B if A c B), and in the truncated Boolean algebra !!l; := !!ln \ {0, [1, n]}. We say that .2' == (Li, L2, ... , Lk) is a k-chain in !!ln if Li E &4n for all 1 ,::::; i ,::::; k and Li is a proper subset of Li+ 1 for all 1 ,::::; i ::; k - 1. A family ff of k-chains in &4n is intersecting if any two elements of ff have non-empty intersection. k-chains and an intersecting family in !!l; are defined analogously. Let f (n, k) and +Research partially supported by NSF Grant CCR-9201303 and NSA Grant MDA904-92-H-3046. §Research partially supported by ONR Grant N-0014-91-J-1385. -, 300 P. L. Erdos, A. Seress and L. A. Szekely f-(n, k) denote the maximum size of intersecting families of k-chains in 98n and :JJ;, respectively. Obviously, the family .~(A) of all k-chains containing some fixed A E f!Bn is an intersecting family, and the same is true for the family ;F-(A) of all k-chains in :!4; containing some fixed A E :Jt;. Our main result is the following. Theorem 1.1. For any k, n, we have ( i) f(n, k) = !ff(0JI and (ii) f-(n,k) = i:P-((1})1. Moreover.for 2sk~n+1, the only extremal.families in 9811 are ff(</J) and ff([l,n]). The most well-known proof techniques for the original EKR Theorem are shifting and the kernel method. (For a brief introduction to these methods, see e.g., the survey papers of Frankl [6] and Filredi [8].) The kernel method usually ensures short and easy proofs, but rarely gives the exact range of the result. Shifting gives exact (but perhaps slightly more complicated) proofs. The situation is very similar in our case: Z. Fiiredi (personal communication) showed, using only the kernel method, that for n ~ 6k Ink Theorem 1.1 ( i) holds. In our proof of Theorem 1.1, we use an analogue of the shifting method and obtain a result without any restrictions on the parameters. We remark, however, that to obtain sharp results in the case of t-intersecting families of chains, or the poset obtained by deleting the top m and bottom m levels in 98 11 for some m < n/2, it seems to be necessary to combine the two methods. Hilton-Milner type generalizations are also possible. Moreover, we have a common generalization of the original EKR theorem and Theorem 1.1. We shall return to these problems in a forthcoming paper. Let S(p, q) denote the Stirling numbers of the second kind, i.e. S(p, q) is the number of partitions of a p-element set into q nonempty parts. It is easy to see that lff-({l})l = k !S(n - l, k ), since each ::!' = (L 1, L2 , ... , Lk) E ;F-({ l}) corresponds to an ordered partition (L2 \ L1,L3 \ L2, ... ,Lk \ Lk-1, [1,n] \ Lk) of [2,n]. Similarly, [.Y"(</J)i = (k- l)!S(n+ l,k) = (k - 1) !S(n, k - I)+ k !S(n, k), the two last terms corresponding to the number of k-chains in ff(</J) containing and not containing [1, n], respectively. In the proofs, we shall often use the well-known recursion S(n,k) = S(n -1,k- l) +kS(n -1,k) (see e.g., [9, Chapter l]). In particular, lff(</J)i = (k- l)!S(n+ 1,k). 2. Shifting In this section we begin the proof of Theorem 1.1. We reduce the problem to the examination of so-called compressed sets of chains and prove that these satisfy a strong intersection property. Let ff be a family of pairwise intersecting k-chains from El 11 or 98-;;, and let l s i < j s n be integers. The (i,j) chain-shift Sij(ff) of the family ff is defined as follows. On Intersecting Chains in Boolean Algebras 301 For every k-chain 2 = (Li. ... Lk) E ff let S.·(!!:') = (L' L') h , •1 1, ••• , k , w ere L; = { L1 \ {j} U {i} if j E L1 and it/. L1, L1 otherwise. We say that L; is the shift of L1. Shifting preserves set containment, so s, ·(!!:') is a k-chain. The shifted family Sij(ff) is obtained by the following rule: replace ever~ k-chain !!:' E § by Sij(!I') if and only if (1) Sij(2)-=!= !I' and (2) Sij(!!:') <f. ff. It is clear from the definition that iSij(ff)I = lffl. Moreover, shifting preserves the intersection property. Lemma 2.1. If ff is an intersecting family of k-chains in !!Jn or :14-;;. then SiJ(.F) is also intersecting. Proof. Let !1'1,22 E Sij(.9'); we have to prove that they contain a common element. We distinguish three cases: Case 1: !I' 1, 2 2 E ff. In this case it is obvious that !I' 1 and !!:' 2 intersect. Case 2: !1'1,!1'2 <f. ff. In this case, there are !!:'3,!1'4 E :F such that !!:'1 = S;j(!t'3) and !1'2 = Sij(!l'4). Let M E 23 n !/:'4. Then the shift of M (which may be M itself) is a common element of !!:' 1 and !!:' 2. Case 3: 21 <f. ff and !!:'2 E !F. Then let !/3 E !ffl such that !t'1 = S;i(Y3). There may be two reasons why 22 was not replaced. If !!:'2 = Sij(.!t'2) then let M E .!t'2 n Y 3. The shift of Mis itself (since .!t'2 = Sij(22 )) so ME !!:'2 n Sij(Y3) = Y 2 n !t'1 as well. The other reason is that !!:'2 -=!= S;j(.!t'2) but Sij(!t'2) E :F. In this subcase, let M E !!:'3 n SiJ(22). It is impossible that j E M and i ff. M since M is the shift of some element of !!:'2. Also, it is impossible that i E M and j <f. M because there is some K E .!t'3 such that j EK and i <f. K (because Sij(.!t'3 ) f !f3) and one of K, M must contain the other. So M is a set containing either both of i,j or neither of i,j. In either case, from M E SiJ(!t'2) we have ME .!t'2 so M E !!:'1 n .!t'2. D We say that the family :F of intersecting k-chains is compressed if § is invariant for all chain-shift operations Sij, 1 ::;; i < j::;; n. By Lemma 2.1, for any intersecting family ff, repeated applications of chain-shifts result in a compressed family of the same size. Compressed families satisfy a strong intersection property. We say that M E id11 (or M E f!J-;;) is an initial segment if M = [1, m] for some 1 ::;; m :s;; n or M = 0. Lemma 2.2. Let ff be a compressed family of intersecting k-chains. Then for any .!t'1, Y2 E !ffl, !1'1 and !t'2 intersect in an initial segment. Proof. Suppose that the lemma is not true and let .!t'1 E :F be a minimal counterexample in the sense that (i) there exists !t'2 E ff such that .Y1 n .Y2 contains no initial segment 302 P. L. Erdos, A. Seress and L. A. Szekely (ii) LLE..'l"i LxEL x is minimal among all !!'1 satisfying (i). Let ME !!'1 n .51'2• Since M is not an initial segment, there exist 1 :::::; i < j:::::; n such that i (/:. M and j E M. Then Sij(!l'i) -::f= !!'1, so Sij(!l'i) is not a counterexample. Therefore, there exists an initial segment K E S;j(!l'i) n !!'2• It is impossible that j EK and i tt K, since K is an initial segment. Also, it is impossible that i E K and j tt K, because K, M E !l' 2, so one of them must contain the other. So K is a set containing both of i, j or neither of i,j. In either case K E !!'1, which is a contradiction. D In the next two sections, we prove Theorem 1.1 for !!fin and !!l;;, respectively. By Lemma 2.1, it is enough to consider compressed families. 3. Chains in f!4 n We prove by induction on n that f(n,k) = (k-1) !S(n + 1, k). The base case n = 1 is trivial. Suppose we are done for n - 1 (with all values of k) and let § be a compressed family of chains in P4n. We distinguish two cases: Case I: § contains a chain .ff such that the only initial segment in !l' is [1, n]. Then, since each chain in § must intersect .ff in an initial segment (see Lemma 2.2), all chains contain [1, n] and we are done. Case 2: There is no chain in § such that [1, n] is the only initial segment in the chain. Then delete n from each element of each chain. Each chain is transformed into either a k-chain or a (k- 1 )-chain and so we obtain an intersecting family rck-l of chains of length k - 1 in !!ln-1 and an intersecting family rck of chains of length k in !!ln-1· We claim that each (L1, ... , Lk-d E rck-l can be obtained from :::::; k - 1 chains of§. This is true since we have to add the set L; u { n} to the chain for some 1 :::::; i :::::; k - 1 and add n to the sets L;+ 1,. .. , Lk-I· The value of i uniquely determines the chain in §. Furthermore, i = 0 is impossible, since then the only initial segment would be [1, n]. We also claim that each (L1,. .. ,Lk) Erck can be obtained from:::::; k chains of :F. Indeed, we have to add n to the sets starting at some 2 ::::;; i:::::; k + 1; the value k + 1 corresponds to the case that n did not occur in any element of the chain in :F. Furthermore, i = 1 is impossible, since the only initial segment would be [1, n]. Thus I/FI 5;, (k - l)f(n - 1,k - 1) + kf(n - 1,k) = (k - 1) !S(n + 1,k). ( 1) The uniqueness of the extremal systems can also be proved by induction on n. First, we remark that if k = n + 1, every family § of k-chains must contain the empty set, and maximality implies§= :7(0). If k = 2 and I/FI ~ 4, then§ s;;; §(A) for some subset A. Now, lff(A)I = 21AI + 2n-IAI - 2, which takes its maximum value for IAI = 0 and IAI = n. In the case 3 :::::; k :::::; n, we first consider a compressed family §. If§ belongs to Case 1 above, then§= :F([l,n]); otherw~se, in Case 2, we must have equality in (1). This implies that rck-1 and rck are extremal families in P4n-i. and, by the induction hypothesis, they must be the :7(0) of (k -1)-chains and k-chains in P4n_ 1, respectively. So§ must be identical with :7(0) in !!ln. On Intersecting Chains in Boolean Algchras 303 Finally, we observe that any family whose compressed image is §'(l)l or .F([L nj) is itself one of these families. ~ We remark that in a preliminary version of the present paper [4], we proved Theorem l.1 for n > klnk. Since then, Ahlswede and Cai [1] have also found a proof for Theorem l.l (i), but their method does not seem to generalize to the truncated case. 4. Chains in .<~;; Again, we use induction on n to prove that f-(n,k) = k!S(11- 1,k). The base case n = 2 is trivial. Suppose we are done for n - 1 and let ff be a compressed family of chains in .'!.a;. We distinguish two cases: Case 1: If there exists a chain !f E .F such that n - 1 E L 1, then !!' may contain only one initial segment, namely [l, n - l]. Then, since each chain in .F must intersect fl' in an initial segment (see Lemma 2.2), all chains contain [l,11- I] and we are done. Case 2: If each !f E ff has no 11- 1 ~ L 1, then, in particular, we never have L1 =!= { n - l). Define ff; {!f E ff: L;+1 - L; = {n -1}}, (i = 1,2, ... ,k- l) ffk {!!' E ff: Lk = {1,2, ... ,n -2,n}}, = :Z uk -:.--r .;r 0 ,J - j=l '7" j· Deleting n-1 from each element of each chain of ff 0, we obtain a family ff~1 of intersecting k-chains in the truncated Boolean algebra on the underlying set { 1, 2, ... , 11 - 2, n }. By hypothesis, 1-Fbi:::::; l-(n - 1,k). Each (L 1, .•• ,Lk) E .'Fb can be obtained from:::::; k chains of ff o, since n - 1 could have been inserted starting at L2, L3, .. ., Lk. or could have been an element of [1, n] \ Lk. Deleting n -1 from every set in every chain in ff; (for any i = 1, 2, ... , k-1 ), we obtain a family .F; of intersecting (k-1 )-chains in the truncated Boolean algebra on the underlying set {1,2, .. .,n - 2,n}. By hypothesis, !.F;! = lff;I:::::; f-(11- 1,k - 1). Finally, define ffk by deleting the largest set Lk = { 1, 2, ... , n - 2, n} from every chain in ff k. Observe that ff~ is a family of intersecting (k - 1 )-chains in the truncated Boolean algebra on the underlying set { l, 2, ... , n - 2, 11 }, since the set that we dropped is not an initial segment in the original underlying set. Therefore, by hypothesis, l..F k I = 1.~~ I ::;; l-(n - 1,k - 1). Hence, !."FI :::::; k · k !S(n - 2, k) + (k - l )(k - 1) !S(11 - 2, k - 1) + (k - l) !S(n - 2, k - l) = k !S(n - l,k). This finishes the proof of Theorem 1.1. D We remark that, analogously to the discussion at the end of Section 3, it can be shown that the only compressed extremal families in £!,B~- are ff--([1]) and ff--w. n - l]). The extension that the only extremal families are ff-(A) with !Al = 1 or !Al = 11 - 1 is still missing. 304 P. L. Erd6s, A. Seress and L. A. Szekely Acknowledgements We are indebted to Ulrich Faigle, Zoltan Furedi, and Walter Kern for stimulating conversations on the subject of the paper. Note added in proof We have just learned of a research program initiated by Miklos Simonovits and Vera T. Sos on 'structured intersection theorems' [10, 11], which has a fairly large literature. They studied the maximum number of graphs on n vertices such that any two intersect in a prescribed graph, e.g. a path or cycle. The following problem fits into their scheme: given a graph G what is the maximum number of pairwise intersecting complete k-subgraphs. In this paper we have studied the comparison graphs of some partially ordered sets. References [l] Ahlswede, R. and Cai, N. (1993) Incomparability and intersection properties of Boolean interval lattices and chain posets, preprint. [2] Deza, M. and Frankl, P. (1983) Erdos-Ko-Rado theorem - 22 years later, SIAM J. Alg. Disc. Methods 4, 419--431. [3] Erdos, P. L., Faigle, U. and Kern, W. ( 1992) A group-theoretic setting for some intersecting Sperner families, Combinatorics, Probabilitv and Computing l, 323-334. [4] Erdos, P. L., Seress. A. and Szekely, L. A. ( 1993) 011 intersecting k-chains in Boolean algebras, Preprint, April 1993. [5] Erdos, P., Ko, C. and Rado, R. ( 1961) Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. 2 12, 313--318. [6] Frankl, P. (1987) The shifting technique in extremal set theory, In: Whitehead, C. (ed.) Surveys in Combinatorics 1987, Cambridge University Press, 81-110. [7] Frankl, P. and Fiiredi, Z. ( 1980) The Erdos-Ko-Rado theorem for integer sequences, SIAM J. Alg. Disc. Methods I, 376-381. [8] Fiiredi, Z. (1991) Turan type problems, In: Keedwell, A. D. (ed.) Surveys in Combinatorics 1991, Cambridge University Press 253-300. [9] Lovasz, L. ( 1977) Combinatorial Problems and Exercises, Akademiai Kiad6, Budapest and North-Holland, Amsterdam. [10] Simonovits, M. and Sos, V. T., Intersection theorems for graphs. Problemes Combinatoires et Theorie des Graphes, Coll. Internationaux C.N.R.S. 260 389-391. [11] Simonovits, M. and Sos, V. T. (1978) Intersection theorems for graphs II. Comhinatorics. Coll. Math. Soc. J. Bolyai 18 1017-1030. 

On intersecting chains in Boolean algebras

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