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Martinsson-Steiner Conjecture on Fractional Chromatic Number

Combinatorics · posed by Anders Martinsson, Raphael Steiner · partial

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Statement

Is the fractional chromatic number of every dd-degenerate triangle-free graph at most (1+o(1))dlog⁡d(1+o(1))\frac{d}{\log d}, with a matching lower bound, as conjectured by Martinsson and Steiner? The upper bound is confirmed constructively for graphs of girth at least 55, and the conjectured lower bound is established in a stronger form for every fixed girth; the original triangle-free case remains open.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. proof attempt · #1

    Peter Allen, Abhishek Dhawan and Jonathan A. Noel, using ChatGPT 5.5 Pro

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The model solved an optimization problem the authors formulated to determine the correct shape of the fractional clique function, checked and simplified probabilistic and algebraic estimates, helped draft some calculations, and pointed the authors to a key reference. The construction and overall strategy are the authors', who take full responsibility.

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