For integers $n\ge k\ge 2$, let $V$ be an $n$-element set, and let $\left(\genfrac{}{}{0ex}{}{V}{k}\right)$ denote the set of all $k$-element subsets of $V$. For disjoint $A,B\subseteq V$, we say $\{A,B\}$ covers $K\in \left(\genfrac{}{}{0ex}{}{V}{k}\right)$ if $K\subseteq A\dot{\cup }B$ and $K$ meets each of $A$ and $B$, i.e., $K\cap A\ne 0̸\ne K\cap B$. We say that a collection $\mathcal{C}$ of such pairs $\{A,B\}$ covers $\left(\genfrac{}{}{0ex}{}{V}{k}\right)$ if every element of $\left(\genfrac{}{}{0ex}{}{V}{k}\right)$ is covered by at least one member of $\mathcal{C}$. When $k=2$, such a family is called a separating system of $V$, where this concept was introduced by Rényi (1961) and studied by many authors.

Let $h(n,k)$ denote the minimum value of ${\sum }_{\{A,B\}\in \mathcal{C}}(\left|A\right|+\left|B\right|)$ among all covers $\mathcal{C}$ of $\left(\genfrac{}{}{0ex}{}{V}{k}\right)$. Hansel (1964) determined the bounds $\lceil n{log}_{2}n\rceil \le h(n,2)\le n\lceil {log}_{2}n\rceil$, and Bollobás and Scott (2007) determined an exact formula for $h(n,2)$. We extend these results to give an exact formula for $h(n,k)$, and to guarantee that all optimal covers $\mathcal{C}$ of $\left(\genfrac{}{}{0ex}{}{V}{k}\right)$ share a common degree-sequence. Our proofs follow lines of Bollobás and Scott, together with weight-shifting arguments in a similar vein to some of Motzkin and Straus (1965).

对于整数 $ñ\ge ķ\ge 2$，让 $V$ 豆角，扁豆 $ñ$-元素集，然后让 $\left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$ 表示全部 $ķ$的元素子集 $V$。不相交$\mathrm{一种}，乙\subseteq V$， 我们说 $\{\mathrm{一种}，乙\}$ 盖子 $ķ\in \left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$ 如果 $ķ\subseteq \mathrm{一种}\dot{\cup }乙$ 和 $ķ$ 遇到每个 $\mathrm{一种}$ 和 $乙$，即 $ķ\cap \mathrm{一种}\ne 0̸\ne ķ\cap 乙$。我们说一个集合$\mathcal{C}$ 这样的对 $\{\mathrm{一种}，乙\}$ 盖子 $\left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$ 如果每个元素 $\left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$ 被至少一名成员覆盖 $\mathcal{C}$。什么时候$ķ=2$，这样的家庭称为 $V$，这个概念由Rényi（1961）提出并被许多作者研究。

让 $H（ñ，ķ）$ 表示的最小值 ${\sum }_{\{\mathrm{一种}，乙\}\in \mathcal{C}}（\left|\mathrm{一种}\right|+\left|乙\right|）$ 在所有封面中 $\mathcal{C}$ 的 $\left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$。汉塞尔（1964）确定了界限$\lceil ñ{日志}_2ñ\rceil \le H（ñ，2）\le ñ\lceil {日志}_2ñ\rceil$，而Bollobás和Scott（2007）确定了 $H（ñ，2）$。我们扩展这些结果以给出精确的公式$H（ñ，ķ）$，并确保所有最佳覆盖 $\mathcal{C}$ 的 $\left(\genfrac{}{}{0ex}{}{V}{ķ}\right)$共有一个度数序列 我们的证明遵循Bollobás和Scott的观点，以及与Motzkin和Straus（1965）相似的权重转移论点。