Infinite Sperner's theorem

TL;DR

This paper investigates the growth rate of infinite antichains in the Boolean lattice, establishing bounds and constructing examples that nearly match these bounds, extending Sperner's theorem to the infinite case.

Contribution

It introduces the first bounds on the growth of infinite antichains and constructs examples that nearly achieve these bounds, extending classical Sperner's theorem.

Findings

Infinite antichains are thinner than finite ones, with growth rate bounded by 2^n/(n log n).

Constructed antichains match the lower bounds up to a constant factor.

Main result shows the bounds are essentially tight, confirming the growth rate limits.

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 $\Theta\big(\frac{2^n}{\sqrt{n}}\big)$. 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 $C>0$.