Abstract: Let m, n, s₁, s₂, …, s_n, be non-negative integers with 0 ≤ m ≤ n. Assume μ(s₁, s₂, …, s_n)={(a₁, a₂, …,a_n)|0 ≤ a_i ≤ s_i for each i} is a poser, Where (a₁, a₂, …,a_n)<(b₁, b₂, …, b_n) if and only if a_i<b_i for all i. A subset J of μ(s₁, s₂, …, s_n) is called a two-part Sperner family in μ(s₁, s₂, …, s_n) if for any a=(a₁, a₂, …,a_n), b=(b₁, b₂, …, b_n) ∈μ(s₁, s₂, …, s_n), (i) a_i=b_i(1 ≤ i ≤ m) and a_i ≤ b_i(m+1 ≤ i ≤ n) imply ai=bi for all i,and (ⅱ) a_i ≤ b_i(1 ≤ i ≤ m)and a_i=b_i(m+1 ≤ i ≤ n) imply a_i=b_i for all i.
In this paper, we prove that if J is a two-part Sperner family in μ(s₁, s₂,…, s_n), then|J| ≤ |{(x₁,x₂,…,x_n)|(x₁,x₂,…,x_n)∈μ,∑_i=1ⁿx_i=[(∑_i=1ⁿS_i)/2]}.