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GENERALIZED TWO-PART SPERNER FAMILIES
Wu Shiquan
Acta mathematica scientia,Series B. 1993, 13 (2):
202-206.
Let m, n, s1, s2, …, sn, be non-negative integers with 0 ≤ m ≤ n. Assume μ(s1, s2, …, sn)={(a1, a2, …,an)|0 ≤ ai ≤ si for each i} is a poser, Where (a1, a2, …,an)<(b1, b2, …, bn) if and only if ai<bi for all i. A subset J of μ(s1, s2, …, sn) is called a two-part Sperner family in μ(s1, s2, …, sn) if for any a=(a1, a2, …,an), b=(b1, b2, …, bn) ∈μ(s1, s2, …, sn), (i) ai=bi(1 ≤ i ≤ m) and ai ≤ bi(m+1 ≤ i ≤ n) imply ai=bi for all i,and (ⅱ) ai ≤ bi(1 ≤ i ≤ m)and ai=bi(m+1 ≤ i ≤ n) imply ai=bi for all i. In this paper, we prove that if J is a two-part Sperner family in μ(s1, s2,…, sn), then|J| ≤ |{(x1,x2,…,xn)|(x1,x2,…,xn)∈μ,∑i=1nxi=[(∑i=1nSi)/2]}.
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