An L-close Sperner family on an n-element set has at most sum_{h=0}^{|L|} C(n,h) members, and at most n members when L contains exactly one positive integer.
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On $L$-close Sperner systems
An L-close Sperner family on an n-element set has at most sum_{h=0}^{|L|} C(n,h) members, and at most n members when L contains exactly one positive integer.