LAUSR

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.