For a subfamily ${F}\subseteq 2^{[n]}$ of the Boolean lattice, consider the graph $G_{F}$ on ${F}$ based on the pairwise inclusion relations among its members. Given a positive integer $t$, how large can ${F}$ be before $G_{F}$ must contain some component of order greater than $t$? For $t=1$, this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For $t=2^n$, this question is trivial. We are interested in what happens between these two extremes. For $t=2^{g}$ with $g=g(n)$ being any integer function that satisfies $g(n)=o(n/\log n)$ as $n\to\infty$, we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer. This constitutes a nontrivial generalisation of Sperner's theorem. We do so by a reduction to a Tur\'an-type problem for rainbow cycles in properly edge-coloured graphs. Among other results, we also give a sharp answer to the question, how large can ${F}$ be before $G_{F}$ must be connected?

翻译：对于布尔格 $2^{[n]}$ 的子族 ${F}\subseteq 2^{[n]}$，考虑基于其成员间两两包含关系定义的图 $G_{F}$。给定正整数 $t$，在 $G_{F}$ 必须包含阶数大于 $t$ 的连通分支之前，${F}$ 的规模最大能是多少？对于 $t=1$，这个问题几乎在一个世纪前已由 Sperner 精确解答：即布尔格中间层的规模。对于 $t=2^n$，该问题是平凡的。我们关注的是这两个极端情况之间的中间情形。对于 $t=2^{g}$，其中 $g=g(n)$ 是满足 $g(n)=o(n/\log n)$（当 $n\to\infty$ 时）的任意整数函数，我们对上述问题给出了渐近精确的答案：其规模不会比中间层的规模大太多。这构成了 Sperner 定理的一个非平凡推广。我们通过将问题归约为恰当边着色图中彩虹圈的 Turán 型问题来实现这一目标。在其他结果中，我们还对“在 $G_{F}$ 必须连通之前，${F}$ 的规模最大能是多少？”这一问题给出了精确解答。