论文标题
无限的Sperner定理
Infinite Sperner's theorem
论文作者
论文摘要
极端集合理论中最古典的结果之一是Sperner的定理，该定理说，布尔晶格中最大的抗抗小节$ 2^{[n]} $具有尺寸$θ\ big（\ frac {2^n} {\ sqrt {n}}}} \ big）$。由Erds的旧问题促进了无限SIDON序列的生长，在本说明中，我们研究了最大无限敌抗的增长率。使用众所周知的Kraft不等式用于前缀代码，并不难证明无限的抗小选应该比相应的有限有限的敌人“薄”。更确切地说，如果$ \ m athcal {f} \ subset 2^{\ mathbb {n}} $是反小节，那么 $ \ liminf_ {n \ rightarrow \ infty} \ big | \ mathcal {f} \ cap 2^{[n]} \ big | \ left（\ frac {2^n} {n} {n} {n} {n} $ \ MATHCAL {f} $使得$ \ liminf_ {n \ rightarrow \ infty} \ big | \ nathcal {f} \ cap 2^{[n]} \ big | | \ left（\ frac {\ frac {2^n} $ C> 0 $。
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 $Θ\big(\frac{2^n}{\sqrt{n}}\big)$. 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 $$\liminf_{n\rightarrow \infty}\big|\mathcal{F} \cap 2^{[n]}\big|\left(\frac{2^n}{n\log n}\right)^{-1}=0.$$ 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 $C>0$.
