Collections of Subsets with the Sperner Property

Let X = {1,. . ., n) and Í" = {1,..., k), k < n. Let C(n, k) be the subsets of X which intersect Y, ordered by inclusion. Lih showed that C(n, k) has the Sperner property. Here it is shown that C(n, k) has several stronger properties. A nested chain decomposition is constructed for C(n, k) by bracketing. C(n, k) is shown to have the LYM property. A more general class of collections of subsets is studied: Let X be partitioned into parts Xx,. . ., Xm, let Ix, . . . , Im be subsets of {0, 1.«}, and let P {Z c X\ \Z n X¡\ e /„ 1 < i < m). Sufficient conditions on the /, are given for P to be LYM, or at least Sperner, and examples are provided in which P is not Sperner. Other results related to Spemer's theorem, the Kruskal-Katona theorem, and the LYM inequality are presented. In 1928 Sperner [18] showed that if S is a collection of subsets of X = {1,.. ., h) such that no set in S contains any other, then This bound is best-possible and is attained if and only if S consists of all subsets of A of size I \n I or all subsets of size \\n\. ([aj and [a] are, respectively, the greatest integer < a and the least integer > a.) Lih [15] recently discovered this generalization of Sperner's theorem: With A as above, let y be a fc-element subset of A, say Y = {I, . . ., k}. Again let 'S be a collection of subsets of A with no set in S containing any other, but with the additional property that each set in S has a nonempty intersection with Y. Then Equality is attained here for example if S consists of all subsets of A of size i^nl which intersect Y. Let C(n, k) denote the collection of all subsets of A which intersect Y, ordered by inclusion. Lih's result is that C(n, k) has the Sperner property. Following a review of the terminology of extremal set theory in §1, we show in §2 that C(n, k) has a stronger property: It is a nested chain order. In §3 we show that C(n, k) has the LYM property, which is also stronger than the Sperner property. We broaden this by looking at collections of all subsets of A with specified intersection sizes with each part of a partition of A. Specifically, Received by the editors August 25, 1980 and, in revised form, January 20, 1981. 1980 Mathematics Subject Classification Primary 05A05; Secondary 06A10. © 1982 American Mathematical Society 0002-9947/82/0000-0775/$05.2S 575 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use