Let \([n]=\{1,2,\dots , n\}\). Let \(\left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) be the family of all subsets of [n] of size k. Define a real-valued weight function w on the set \(\left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) such that \(\sum _{X\in \left( {\begin{array}{c}[n]\\ k\end{array}}\right) } w(X)\ge 0\). Let \({\mathcal {U}}_{n,t,k}\) be the set of all \({\mathbf {P}}=\{P_1,P_2,\dots ,P_t\}\) such that \(P_i\in \left( {\begin{array}{c}[n]\\ k\end{array}}\right) \) for all i and \(P_i\cap P_j=\varnothing \) for \(i\ne j\). For each \({\mathbf {P}}\in \mathcal {U}_{n,t,k}\), let \(w({\mathbf {P}})=\sum _{P\in {\mathbf {P}}} w(P)\). Let \(\mathcal {U}_{n,t,k}^+(w)\) be set of all \({\mathbf {P}}\in \mathcal {U}_{n,t,k}\) with \(w({\mathbf {P}})\ge 0\). In this paper, we show that \(\vert \mathcal {U}_{n,t,k}^+(w)\vert \ge \frac{\prod _{1\le i\le (t-1)k} (n-tk+i)}{(k!)^{t-1}((t-1)!)}\) for sufficiently large n.

让\([n]=\{1,2,\dots , n\}\)。令\(\left( {\begin{array}{c}[n]\\ k\end{array}}\right) \)是大小为k的 [ n ] 的所有子集的族。在集合\(\left( {\begin{array}{c}[n]\\ k\end{array}}\right) \)上定义一个实值权重函数w使得\(\sum _{ X\in \left( {\begin{array}{c}[n]\\ k\end{array}}\right) } w(X)\ge 0\)。让\（{\ mathcal【U}} _ {N，T，K} \）是集合中的所有的\（{\ mathbf {P}} = \ {P_1，P_2，\点，P_T \} \）这样即\（P_I \在\左（{\开始{阵列} {C} [n]的\\ķ\ {端阵列}} \右）\）对于所有我和\（P_I \帽P_j = \ varnothing \）对于\(i\ne j\)。对于每个\({\mathbf {P}}\in \mathcal {U}_{n,t,k}\)，让\(w({\mathbf {P}})=\sum _{P\in {\mathbf {P}}} w(P)\)。让\(\mathcal {U}_{n,t,k}^+(w)\)设置为所有\({\mathbf {P}}\in \mathcal {U}_{n,t,k }\)与\(w({\mathbf {P}})\ge 0\)。在本文中，我们证明\(\vert \mathcal {U}_{n,t,k}^+(w)\vert \ge \frac{\prod _{1\le i\le (t-1 )k} (n-tk+i)}{(k!)^{t-1}((t-1)!)}\)对于足够大的n。