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: $$(i)p\mathrm{\mid }F\setminus G\mathrm{\mid }=q\mathrm{\mid }G\setminus F\mathrm{\mid },\text{ and}$$ $$(ii)f>g\text{ for all }f\in F\setminus G\text{ and }g\in G\setminus F\mathrm{.}$$ Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family with patterns on $[n]$ is $$O(e^{120\sqrt{\log n}}\frac{2^n}{\sqrt{n}})\mathrm{.}$$ We improve and generalize this result, and prove that the cardinality of every ($p,q$)-tilted Sperner family with patterns on [$n$] is $$O(\sqrt{\log n}\frac{2^n}{\sqrt{n}})\mathrm{.}$$