A note on tilted Sperner families with patterns

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 $O(\sqrt{\log n} \ \frac{2^n}{\sqrt{n}}).$

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