  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 $\Theta\big(\frac{2^n}{\sqrt{n}}\big)$. Motivated by an old problem of
Erd\H{o}s 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
  $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap
2^{[n]}\big|\left(\frac{2^n}{n\log n}\right)^{-1}=0.$$ Our main result shows
that this bound is essentially tight, that is, we construct an antichain
$\mathcal{F}$ such that $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap
2^{[n]}\big|\left(\frac{2^n}{n\log^{C} n}\right)^{-1}>0$$ holds for some
absolute constant $C>0$.
