Lu–Thompson conjecture on off-diagonal Boolean-lattice Ramsey numbers

Let $Q_m$ and $Q_n$ be Boolean lattices of dimensions $m$ and $n$, respectively, with $n\ge m$. Let $R(Q_m,Q_n)$ be the least dimension of a Boolean lattice such that every blue/red coloring contains a blue copy of $Q_m$ or a red copy of $Q_n$.

Lu–Thompson conjecture.

$$R(Q_m,Q_n)=o(n^2).$$

The source presents this as an improvement over the basic quadratic upper bound and notes that the fixed-$m$ case is already linear. The conjecture concerns the regime in which both dimensions are large.