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On maximal antichains containing no set and its complement

✍ Scribed by G.F. Clements; H.-D.O.F. Gronau

Publisher
Elsevier Science
Year
1981
Tongue
English
Weight
1016 KB
Volume
33
Category
Article
ISSN
0012-365X

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✦ Synopsis

Let 16k,dk,+.. d k,, be integers and let S denote the set of all vectors x =(x1, . . . , x,) with integral coordinates sz@@ing 05% "4, i = 1,2, . . . r n; equivalently, .S is the set of ail subset of' a mtitiset eonsistiitg of & elements of type i, i = 1,2,. . . , n. A subset X of 3" is an antichain if and otiy if for any two erectors x and y in X the inequalities I+ G yiii = 1,2, . . . , n, do not all hold. For an arbitraq subset H of S, [i)H dcnutes the subset of H cunsistlng of vectors with component sum i, i = 0, 1,2,. -. , K, where K = k, + k, + l l * + k,. [$I! denotes the number of vectq in H, and *be c~~ph+ of a vector AXE S js (k,-x1, k2-J%, . . . , k, -x,). What is the ma&& card&&y of 'ti ki~tkhain contaitiing no vector and its complement? The answer is obtained as a corollary of the fQ&wiszg theorem: if X is un anti?tain, K is ~WI and /($K)X/ dws not exceeCt the number of actors in (&)S witir fist coordinate &&rent from k,, then

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