The Kelly--Trotter conjecture and dimension of poset products

We study the order dimension of Cartesian products of finite posets. Kelly and Trotter conjectured in 1982 that $\dim(P\times Q)\ge\dim P+\dim Q-2$ for all finite posets $P$ and $Q$. For $m\ge3$, let $R_m$ denote the incidence poset of the complete graph on $m$ vertices. We prove that there is a con