The Kelly--Trotter product conjecture for posets of dimension three

Abstract

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.

Disclosure

“Use of generative AI. During the preparation of this work, the authors used generative AI to improve the readability and language of the manuscript and to assist with writing the Python verification program for the ten fixed posets. The authors carefully reviewed and verified the manuscript and take full responsibility for its”

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Classification

Code generation, completion, or debugging

Multiplier

2

Verified

2026-08-23T05:15:28.830678Z

Structural counts