A note on infinite antichain density

Abstract

Let $\mathcal{F}$ be an antichain of finite subsets of $\mathbb{N}$. How quickly can the quantities $|\mathcal{F}\cap 2^{[n]}|$ grow as $n\to\infty$? We show that for any sequence $(f_n)_{n\ge n_0}$ of positive integers satisfying $\sum_{n=n_0}^\infty f_n/2^n \le 1/4$, $f_{n_0}=1$ and $f_n\le f_{n+1}\le 2f_n$, there exists an infinite antichain $\mathcal{F}$ of finite subsets of $\mathbb{N}$ such that $|\mathcal{F}\cap 2^{[n]}| \geq f_n$ for all $n\ge n_0$. It follows that for any $\varepsilon>0$ there exists an antichain $\mathcal{F}\subseteq 2^\mathbb{N}$ such that $$\liminf_{n \to \infty} |\mathcal{F}\cap 2^{[n]}| \cdot \left(\frac{2^n}{n\log^{1+\varepsilon} n}\right)^{-1} > 0.$$ This resolves a problem of Sudakov, Tomon and Wagner in a strong form, and is essentially tight.