AI 中文总结
该研究证明了三维偏序集下的Kelly–Trotter乘积猜想，否定了Trotter的相关猜想，还得出特定偏序集乘积维数的结论，所用方法涉及偏序集分类与图着色搜索。
AI 中文摘要
Kelly和Trotter提出猜想：对所有有限偏序集P和Q，有dim(P×Q)≥dim P + dim Q - 2。我们证明了当dim P=dim Q=3时该猜想成立，同时否定了Trotter的另一猜想：对每个1≤m≤n，存在有限偏序集P和Q使得dim P=m、dim Q=n且dim(P×Q)=n。我们还证明，对每个dim P=3的有限偏序集P和每个k≥3的 crown C_k，都有dim(C_k×P)=4。证明用到了3-不可约偏序集和临界对图的分类：对6个无限非crown族，我们构造了显式的非3-可着色子图；对10个固定偏序集，通过穷尽式3-着色搜索处理。
英文摘要
Kelly and Trotter conjectured that dim(P x Q) >= dim P + dim Q - 2 for all finite posets P and Q. We prove the conjecture when dim P = dim Q = 3. This also disproves Trotter's conjecture that, for every 1 <= m <= n, there exist finite posets P and Q with dim P = m, dim Q = n, and dim(P x Q) = n. We further prove that dim(C_k x P) = 4 for every finite poset P with dim P = 3 and every crown C_k with k >= 3. The proof uses the classification of 3-irreducible posets and graphs of critical pairs. For the six infinite noncrown families, we construct explicit non-3-colorable subgraphs. The ten fixed posets are handled by an exhaustive 3-coloring search.
