Kelly–Trotter product conjecture for posets

Proposed in 1982, the Kelly–Trotter conjecture asks whether finite posets satisfy $\dim(P\times Q)\ge \dim P+\dim Q-2$.

Known results

• Reuter proved the conjecture for pairs of standard posets $S_m,S_n$ and for $P=Q$ when $\dim P=3$.
• Reuter established $\dim(S_n\times S_n)=2n-2$.
• General bounds remain $\max(\dim P,\dim Q)\le\dim(P\times Q)\le\dim P+\dim Q$.

August 2026: dimension-three theorem

Zhaochen Dong and Kaiyun Wang’s preprint claims that $\dim(P)=\dim(Q)=3$ implies $\dim(P\times Q)\ge4$, proving the conjectured bound in the $3\times3$ case. It also rules out Trotter’s proposed examples there and claims $\dim(C_k\times P)=4$ for $k\ge3$ and $\dim(P)=3$.

Current status (as of August 2026): The $3\times3$ case is established by the new preprint, while the general inequality remains open.