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Macaulay posets are posets for which there is an analogue of the classical KruskalKatona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets, where the intention is to present them as pieces of a general theory. In(More)
It is proved that, for any positive integer m, the weight of the unionclosure of any m distinct 2-sets is at least as large as the weight of the union-closure of the first m 2-sets in squashed (antilexicographic) order, where all i-sets have the same non-negative weight wi with wi ≤ wi+1 for all i, and the weight of a family of sets is the sum of the(More)
We study maximal families A of subsets of [n] = {1, 2, . . . , n} such that A contains only pairs and triples and A 6⊆ B for all {A,B} ⊆ A, i.e. A is an antichain. For any n, all such families A of minimum size are determined. This is equivalent to finding all graphs G = (V,E) with |V | = n and with the property that every edge is contained in some triangle(More)
We consider the poset SO(n) of all words over an n{element alphabet ordered by the subword relation. It is known that SO(2) falls into the class of Macaulay posets, i.e. there is a theorem of Kruskal{Katona type for SO(2). As the corresponding linear ordering of the elements of SO(2) the vip{order can be chosen. Daykin introduced the V {order which(More)
A ranked poset P is a Macaulay poset if there is a linear order of the elements of P such that for any m; i the set C(m; i) of the m (with respect to) smallest elements of rank i has minimum{sized shadow among all m{element subsets of the i{th level, and the shadow of C(m; i) consists of the smallest elements of the (i ? 1){st level. P is called(More)
Some inequalities for cross-unions of families of finite sets are proved that are related to the problem of minimizing the unionclosure of a uniform family of given size. The cross-union of two familiesF andG of subsets of [n] = {1, 2, . . . , n} is the familyF ∨ G = {F ∪ G : F ∈ F ,G ∈ G}. It is shown that |F ∨ G|/|F | ≥ |G ∨ Bn|/2, where Bn denotes the(More)