[2608.14434] The Kelly--Trotter product conjecture for posets of dimension three Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.14434 (math) [Submitted on 14 Aug 2026] Title: The Kelly--Trotter product conjecture for posets of dimension three Authors: Zhaochen Dong , Kaiyun Wang View a PDF of the paper titled The Kelly--Trotter product conjecture for posets of dimension three, by Zhaochen Dong and 1 other authors View PDF HTML (experimental) 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. Subjects: Combinatorics (math.CO) MSC classes: 06A07, 05C15 Cite as: arXiv:2608.14434 [math.CO] (or arXiv:2608.14434v1 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.14434 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Kaiyun Wang [ view email ] [v1] Fri, 14 Aug 2026 16:16:32 UTC (19 KB) Full-text links: Access Paper: View a PDF of the paper titled The Kelly--Trotter product conjecture for posets of dimension three, by Zhaochen Dong and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from