A Generalization of Sperner's Theorem on Compressed Ideals

Let $[n]=\{1,2,\ldots,n\}$ and $\mathscr{B}_n=\{A: A\subseteq [n]\}$. A family $\mathscr{A}\subseteq \mathscr{B}_n$ is a Sperner family if $A\nsubseteq B$ and $B\nsubseteq A$ for distinct $A,B\in\mathscr{A}$. Sperner's theorem states that the density of the largest Sperner family in $\mathscr{B}_n$ is $\binom{n}{\left\lceil{n/2}\right\rceil}/2^n$. The objective of this note is to show that the same holds if $\mathscr{B}_n$ is replaced by compressed ideals over $[n]$.

Families that Remain $k$-Sperner Even After Omitting an Element of their Ground Set

A family $\mathcal{F}\subseteq 2^{[n]}$ of sets is said to be $l$-trace $k$-Sperner if for any $l$-subset $L \subset [n]$ the family $\mathcal{F}|_L=\{F|_L:F \in \mathcal{F}\}=\{F \cap L: F \in \mathcal{F}\}$ is $k$-Sperner, i.e. does not contain any chain of length $k+1$. The maximum size that an $l$-trace $k$-Sperner family $\mathcal{F} \subseteq 2^{[n]}$ can have is denoted by $f(n,k,l)$.For pairs of integers $l<k$, if in a family $\mathcal{G}$ every pair of sets satisfies $||G_1|-|G_2||<k-l$, then $\mathcal{G}$ possesses the $(n-l)$-trace $k$-Sperner property. Among such families, the largest one is $\mathcal{F}_0=\{F\in 2^{[n]}: \lfloor \frac{n-(k-l)}{2}\rfloor+1 \le |F| \le \lfloor \frac{n-(k-l)}{2}\rfloor +k-l\}$ and also $\mathcal{F}'_0=\{F\in 2^{[n]}: \lfloor \frac{n-(k-l)}{2}\rfloor \le |F| \le \lfloor \frac{n-(k-l)}{2}\rfloor +k-l-1\}$ if $n-(k-l)$ is even.In an earlier paper, we proved that this is asymptotically optimal for all pair of integers $l<k$, i.e. $f(n,k,n-l)=(1+o(1))|\mathcal{F}_0|$. In this paper we consider the case when $l=1$, $k\ge 2$, and prove that $f(n,k,n-1)=|\mathcal{F}_0|$ provided $n$ is large enough. We also prove that the unique $(n-1)$-trace $k$-Sperner family with size $f(n,k,n-1)$ is $\mathcal{F}_0$ and also $\mathcal{F}'_0$ when $n+k$ is odd.

Quadratic LYM-type inequalities for intersecting Sperner families

International audience Let $\mathcal{F}\subseteq 2^{[n]}$ be a intersecting Sperner family (i.e. $A \not\subset B, A \cap B \neq \emptyset$ for all $A,B \in \mathcal{F}$) with profile vector $(f_i)_{i=0 \ldots n}$ (i.e. $f_i=|\mathcal{F} \cap \binom{[n]}{i}|$). We present quadratic inequalities in the $f_i$'s which sharpen the previously known linear $\mathrm{LYM}$-type inequalities.