A note on tilted Sperner families with patterns

TL;DR

This paper generalizes and improves bounds on the size of tilted Sperner families with patterns, showing they are at most on the order of rac{rac{2^n}{\u00a0 ext{sqrt}(n)}}{ ext{rac{ ext{log} n}{ ext{rac{1}{2}}}}} for all p,q.

Contribution

The authors extend previous results by providing a tighter upper bound on the size of (p,q)-tilted Sperner families with patterns.

Findings

Improved the upper bound to O(rac{rac{ ext{log} n}{ ext{rac{1}{2}}}} rac{2^n}{ ext{rac{1}{2}} n}) for all p,q.

Generalized the bound from the specific (1,2) case to all p,q.

Enhanced understanding of the structure and limitations of tilted Sperner families.

Abstract

Let $p$ and $q$ be two nonnegative integers with $p+q>0$ and $n>0$. We call $\mathcal{F} \subset \mathcal{P}([n])$ a \textit{(p,q)-tilted Sperner family with patterns on [n]} if there are no distinct $F,G \in \mathcal{F}$ with: $$(i) \ \ p|F \setminus G|=q|G \setminus F|, \ \textrm{and}$$ $$(ii) \ f > g \ \textrm{for all} \ f \in F \setminus G \ \textrm{and} \ g \in G \setminus F.$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\ \frac{2^n}{\sqrt{n}}).$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$$