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Reading's Problem 9.3: Order Dimension Versus Rank for Simplicial Arrangements

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order-dimension-beyond-rank-simplicial-arrangementsCombinatoricsposed by Nathan Reading; Segovia, 2016recorded: disproved

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Statement

Reading computed the order dimension of the poset of regions for most finite Coxeter arrangements, observed that an exceptional type whose dimension exceeds its rank would be the first known simplicial arrangement with that property, and recorded the general guess that every simplicial region poset has dimension equal to its rank (Problem 9.3 of his 2016 chapter); Segovia later asked the analogous question for oriented-poset lattices. False: the order dimension of the poset of regions can exceed the rank - the Coxeter arrangements H4H_4 and E6E_6 satisfy dim⁡W(H4)≥5\dim W(H_4) \ge 5 and dim⁡W(E6)≥7\dim W(E_6) \ge 7.

Context

A recorded problem of Reading's from the standard reference chapter on posets of regions, open ten years, answered with the first simplicial arrangements whose region posets have dimension above rank. Specialist band at 15.

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  • #1

    Attempt 1

    constructionChatGPT 5.6 Sol Ultra with Daria Poliakova ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    ChatGPT 5.6 Sol Ultra
    people
    Daria Poliakova

    The declaration, in full: "The small obstruction subgraphs were found by ChatGPT 5.6 Sol Ultra. The human input was the belief that the rank guess is incorrect, and one should look for counterexamples." The obstruction subgraphs are the entire content of the disproof, so the model produced the central objects under human direction.

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