Let $n>k>0$ be integers and $X$ an $n$-element set. A family $\mathcal{F}$ consisting of subsets of $X$ is called $k$-Sperner if it has no distinct members $F_0,\dots ,F_k$ such that $F_0\subset F_1\subset \dots \subset F_k$. A family is called $s$-union if the union of any two of its members has size at most $s$. A classical result of Milner determines the maximum size of a family that is both 1-Sperner and $s$-union. The present paper is dealing with the case $k\ge 2$. If $s=2r<n$ then the natural construction is to take all subsets $F\subset X$ with $r-k<\left|F\right|\le r$. Theorem 4.1 shows that this is optimal for $n>r(r+3)$. The case of $s=2r+1$ is more complex. We believe that Example 1.9 provides the maximum. Theorem 1.12 confirms this for $k=2$ and $n\ge r^2+4r+1$.

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if $F\cap G\ne 0̸$ for all $F\in \mathcal{F}$, $G\in \mathcal{G}$. What is the maximum of $\left|\mathcal{F}\right|+\left|\mathcal{G}\right|$ if in addition $\mathcal{F}$ is $k$-Sperner, $\mathcal{G}$ is $\ell$-Sperner? The exact answer is given by Theorem 1.4.

In Section 3 we prove the analogue of Milner’s Theorem for vector spaces.

让 $ñ>ķ>0$ 是整数和 $X$ 一个 $ñ$-元素集。一个家庭$\mathcal{F}$ 由...的子集组成 $X$ 叫做 $ķ$-如果没有独立成员，则为Sperner$F_0，\dots ，F_{ķ}$ 这样 $F_0\subset F_{\mathrm{1个}}\subset \dots \subset F_{ķ}$。一个家庭被称为$s$-工会，如果其任何两个成员的工会的规模最大为 $s$。Milner的经典结果决定了1-Sperner和$s$-联盟。本文件正在处理此案$ķ\ge 2$。如果$s=2\mathrm{[R}<ñ$ 那么自然的构造就是取所有子集 $F\subset X$ 与 $\mathrm{[R}-ķ<\left|F\right|\le \mathrm{[R}$。定理4.1表明这对于$ñ>\mathrm{[R}（\mathrm{[R}+3）$。的情况下$s=2\mathrm{[R}+\mathrm{1个}$更复杂。我们相信，示例1.9提供了最大的限制。定理1.12证实了$ķ=2$ 和 $ñ\ge {\mathrm{[R}}^2+4\mathrm{[R}+\mathrm{1个}$。

两个家庭 $\mathcal{F}$ 和 $\mathcal{G}$ 称为交叉相交 $F\cap G\ne 0̸$ 对所有人 $F\in \mathcal{F}$， $G\in \mathcal{G}$。最大值是多少$\left|\mathcal{F}\right|+\left|\mathcal{G}\right|$ 如果另外 $\mathcal{F}$ 是 $ķ$-Sperner， $\mathcal{G}$ 是 $\ell$-Sperner？确切的答案由定理1.4给出。

在第3节中，我们证明了向量空间的米尔纳定理的类似物。