One of the most classical results in extremal set theory is Sperner's theorem, which says that the largest antichain in the Boolean lattice $2^{[n]}$ has size $\mathrm{\Theta }\left(\frac{2^n}{\sqrt{n}}\right)$. Motivated by an old problem of Erdős on the growth of infinite Sidon sequences, in this note we study the growth rate of maximum infinite antichains. Using the well known Kraft's inequality for prefix codes, it is not difficult to show that infinite antichains should be “thinner” than the corresponding finite ones. More precisely, if $\mathcal{F}\subset 2^{\mathbb{N}}$ is an antichain, then$\underset{n\to \infty }{\lim \inf }|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n\log n}\right)}^{-1}=0.$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $\mathcal{F}$ such that$\underset{n\to \infty }{\lim \inf }|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n{\log }^{C}n}\right)}^{-1}>0$ holds for some absolute constant $C>0$.

极值集理论中最经典的结果之一是 Sperner 定理，它表示布尔格中的最大反链 $2^{[n]}$ 有尺寸 $\mathrm{\Theta }\left(\frac{2^n}{\sqrt{n}}\right)$. 受 Erdős 关于无限 Sidon 序列增长的一个老问题的启发，在本笔记中，我们研究了最大无限反链的增长速度。使用众所周知的前缀码的 Kraft 不等式，不难证明无限反链应该比相应的有限反链“更薄”。更准确地说，如果$\mathcal{F}\subset 2^{\mathbb{N}}$ 是一个反链，那么$\underset{n\to \infty }{\mathrm{林}\mathrm{信息}}|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n\mathrm{日志}n}\right)}^{-1}=0.$ 我们的主要结果表明这个界限本质上是紧的，也就是说，我们构建了一个反链 $\mathcal{F}$ 以至于$\underset{n\to \infty }{\mathrm{林}\mathrm{信息}}|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n{\mathrm{日志}}^{C}n}\right)}^{-1}>0$ 对某个绝对常数成立 $C>0$.