Infinite Sperner's theorem

Abstract

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 $\hat{I}˜\left(\frac{2^n}{\sqrt{n}}\right)$. 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 $$\underset{n\to \infty }{\mathrm{liminf}}|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n\log n}\right)}^{-1}=0.$$ Our main result shows that this bound is essentially tight, that is, we construct an antichain $\mathcal{F}$ such that $$\underset{n\to \infty }{\mathrm{liminf}}|\mathcal{F}\cap 2^{[n]}|{\left(\frac{2^n}{n{\log }^{C}n}\right)}^{-1}>0$$ holds for some absolute constant $C>0$.

9 pages, 1 figure. Dömötör Pálvölgyi brought to our attention that Kraft's inequality for prefix codes can be used to simplify our argument for the upper bound