In this note we give a very short proof for the description of all maximum size two-part Sperner systems

Aydinian, H.

Erdős, Péter

Aydinian H, Erdos PL. All maximum size two-part Sperner systems: In short. Combinatorics, Probability and Computing. 2007;16(4):553-555.In this note we give a very short proof for the description of all maximum size two-part Sperner systems

Aydinian, Haratyun

Erdos, Peter L.

Publications at Bielefeld University

All maximum size two-part Sperner systems: In short

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Combinatorics, Probability and Computing: page 1 of 3. c© 2007 Cambridge University Pressdoi:10.1017/S0963548306007930All Maximum Size Two-PartSperner Systems: In ShortHAROUT AYDINIAN1 and P E´TER L. ERDO˝S2†‡1Department of Mathematics, University of Bielefeld, PO Box 100131, D-33501, Bielefeld, Germany(e-mail: ayd@uni-bielefeld.de)2A. Re´nyi Institute of Mathematics, Hungarian Academy of Sciences,Budapest, PO Box 127, H-1364 Hungary(e-mail: elp@renyi.hu)Received 28 July 2005; revised 18 December 2005In this note we give a very short proof for the description of all maximum size two-partSperner systems.Katona [6] and Kleitman [8] independently observed that the statement of the Spernertheorem remains unchanged if the conditions are relaxed in the following way. LetX = X1 ∪ X2 be a partition of the underlying set X, |Xi| = ni, n1 + n2 = n (with n1  n2).We say that F is a two-part Sperner family if E, F ∈ F, E  F ⇒ ∀i : (F \ E) ⊂ Xi. It wasproved that the size of a two-part Sperner family cannot exceed(nn/2	). It took 20 years toﬁnd all maximum size two-part Sperner families [3]. The description requires some moredeﬁnitions.The two-dimensional proﬁle matrix M(F) is deﬁned by Mij(F) = #{F ∈ F : |X1 ∩F | = i, |X2 ∩ F | = j}. This can be considered as a point in the real space R(n1+1)(n2+1). Theproﬁle matrices of the two-part Sperner families determine a point set in this space, and itis known that the vertices of their polytope are the proﬁle matrices of the homogeneoussystems (see [2, Theorem 3.2]), where a family F is called homogeneous (with respect to thepartition X1, X2) if F ∈ F implies E ∈ F for all sets E satisfying ∀i : |E ∩ Xi| = |F ∩ Xi|.A homogeneous family can be described by the set I(F) = {(i1, i2) : ∀j : |F ∩ Xj | = ij forsome F ∈ F}. If F is a homogeneous two-part Sperner family, then I(F) cannot containpairs with the same ﬁrst or second components, respectively. Consequently we have|I(F)|  n2 + 1. We say that a homogeneous family F is full if |I(F)| = n2 + 1. Then, for† Supported, in part, by the Hungarian NSF, under contract nos. T37846, AT048826.‡ This work was carried out when the second author visited the University of Bielefeld under the Marie CurieFIST Mobility Program.2 H. Aydinian and P. L. Erdo˝severy i2 (0  i2  n2), there is a unique f(i2) such that (f(i2), i2) ∈ I(F). A homogeneousfamily is called well-paired if it is full, {f(i) : i = 0, 1, . . . , n2} is the (n2 + 1)-element intervalaround n1/2	 and (n2i)<(n2j)implies(n1f(i))(n1f(j))(1)for every pair 1  i, j  n2. It is clear that well-pairing is not unique.Theorem 1 ([3]). Let F be a two-part Sperner family with parts X1, X2. Then|F| (n n2	)with equality if and only if F is a homogeneous well-paired family.In [9] and [4] further proofs were given but none were really short or easy to understand.Here we give a very short proof of Theorem 1.Proof. Let F ⊂ X2. Then deﬁne F(F) := {E ⊂ X1 : E ∪ F ∈ F}. Now F(F) is a Spernerfamily, therefore, due to the well-known LYM (Lubell–Yamamoto–Meshalkin) inequality,∑E∈F(F)1(n1|E|)  1, (2)with equality if and only if F(F) consists of a full level in the subset-lattice of X1. (Fordetails see, for example, [7].) Summing this inequality over all i-element subsets of X2, wehave, for all i = 0, . . . , n2, ∑F∈(X2i )∑E∈F(F)1(n1|E|) 1(n2i)  1. (3)Finallyn2∑i=0∑F∈(X2i )∑E∈F(F)1(n1|E|)(n2i) =∑j,iMji(F)(n1j)(n2i)  n2 + 1. (4)The middle term of inequality (4) is a linear function of the proﬁle matrix, and it is easy tocheck that for all full homogeneous families (including well-paired homogeneous families)inequality (4) holds with equality. Therefore all maximum size two-part Sperner familiesmust also satisfy it with equality.Indeed, the cardinality of a family is the sum of the entries in its proﬁle matrix. This isa positive linear function of the proﬁle matrices, and therefore its maximum is attainedonly by points which are on the facets of the polytope spanned by the proﬁle matricesof maximum size homogeneous two-part Sperner families. So the proﬁle matrix of anymaximum size two-part Sperner family is a convex linear combination of proﬁle matricesof maximum size homogeneous families, and hence it satisﬁes any linear equality whichis also satisﬁed by these vertices.Consequently, for maximum size families inequalities (2) and (3) must hold withequality. Therefore, due to (3), each F ⊂ X2 must be a trace on X2 of some element of F.Furthermore, for all F ⊂ X2 the family F(F) must be a full level in the subset-lattice of X1.All Maximum Size Two-Part Sperner Systems: In Short 3We consider ﬁrst the case n1 = n2. In this case we can repeat the reasoning for all E ⊂ X1,which proves that in this case each maximum size two-part Sperner system is homogeneous.In the case of n1 > n2, denote by F(j) ⊂ 2X2 the set of all F ⊂ X2 such that E ∈ F(F)with a ﬁxed E ⊂ X1, |E| = j. (By the previous statement all j-element subsets of X1would deﬁne the same set.) Therefore, for all j ∈ {1, . . . , n1}, F(j) is a Sperner family,and F(j) ∩ F(j ′) = ∅ holds for all j = j ′. Furthermore, any t families among them forma t-Sperner family (their union does not contain a (t+ 1)-chain), and therefore, by PaulErdo˝s’s theorem [1],t∑k=1|F(jk)|is at most the sum of the t largest binomial coeﬃcients. Now, the cardinality of ourmaximum size two-part Sperner system is|F| =n1∑j=0(n1j)|F(j)|. (5)We get the largest value(nn/2)in (5) with the following greedy algorithm. Let B0  B1 · · ·  Bn1 and A0  · · ·  An2 be enumerations of the binomial coeﬃcients(n1j)and(n2i),respectively. If A0 > A1 then we have to match B0 with A0. We have already matched thepairs up to k − 1. Now Ak = Ak+1. Then Bk and Bk+1 must be matched with these twoitems. Recall that the corresponding F(j) and F(j ′) families are Sperner families, andso, by the LYM inequality, they may have that many elements if and only if each of thetwo families is equal to one full level in the subset lattice of X2. So we get that everymaximum size two-part Sperner family must be well-paired.The authors are indebted to the unknown referee for his help.References[1] Erdo˝s, Paul (1945) On a lemma of Littlewood and Oﬀord. Bull. Amer. Math. Soc. 51 898–902.[2] Erdo˝s, P. L. and Katona, G. O. H. (1986) Convex hulls of more-part Sperner families. GraphsCombin. 2 123–134.[3] Erdo˝s, P. L. and Katona, G. O. H. (1986) All maximum 2-part Sperner families. J. Combin.Theory Ser. A 43 58–69.[4] Erdo˝s, P. L., Fu¨redi, Z. and Katona, G. O. H. (2005) Two-part and k-Sperner families: Newproofs using permutations. SIAM J. Discrete Math. 19 489–500.[5] Fu¨redi, Z., Griggs, J. R., Odlyzko, A. M. and Shearer, J. M. (1987) Ramsey–Sperner theory.Discrete Math. 63 143–152.[6] Katona, G. O. H. (1966) On a conjecture of Erdo˝s and a stronger form of Sperner’s theorem.Studia Sci. Math. Hungar. 1 59–63.[7] Katona, G. O. H. (2000) The cycle method and its limits. In Numbers, Information and Complexity(Altho¨fer et al., eds), Kluwer, pp. 129–141.[8] Kleitman, D. J. (1965) On a lemma of Littlewood and Oﬀord on the distribution of certainsums. Math. Z. 90 251–259.[9] Shahriari, S. (1996) On the structure of maximum 2-part Sperner families. Discrete Math. 162229–238.

All maximum size two-part Sperner systems - in short

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