Large subposets with small dimension
  Dorais asked for the maximum guaranteed size of a dimension $d$ subposet of an $n$-element poset. A lower bound of order $\sqrt{n}$ was found by Goodwillie. We provide a sublinear upper bound for each $d$. For $d=2$, our bound is $n^{0.8295}$. 
Benjamin Reiniger and Elyse Yeager
math.CO