Largest component in Boolean sublattices

For a subfamily \(\mathcal{F}\subseteq 2^{[n]}\) of the Boolean lattice, consider the graph \(G_\mathcal{F}\) on \(\mathcal{F}\) based on the pairwise inclusion relations among its members. Given a positive integer t, how large can \(\mathcal{F}\) be before \(G_\mathcal{F}\) 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 = 2ⁿ, 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 Turán-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 \(\mathcal{F}\) be before \(G_\mathcal{F}\)must be connected?