ProbXiv
sign in
Problem archiveProblem record

Log-Concavity of Flats of Matroids

Combinatorics · posed by J. H. Mason, 1972 · disproved

1 comment

Statement

Mason conjectured the following: let MM be a matroid of rank rr, and let WiW_i denote the number of flats of MM of rank ii. Is it true that for all 1≤i≤r−11 \leq i \leq r - 1, we have Wi2≥Wi+1Wi−1W_i^2 \geq W_{i + 1}W_{i - 1}? This is false; a counterexample is given by a graphic matroid whose graph is a generalized theta graph with 7979 edges.

Record

Source
Added
Links

Comments

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

  1. construction · #1

    ChatGPT-5.5 Pro, Claude Opus 4.8, with Matt Larson

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    The author prompted ChatGPT-5.5 Pro to search for counterexamples to Mason's conjecture. After finding none on at most 99 elements, the author expanded his search to consider matroids on large ground sets realizable over F5\mathbb{F}_5 and at failures of log-concavity at high indices. ChatGPT-5.5 Pro found a variant of the given counterexample.

    ChatGPT-5.5 Pro and Claude Opus 4.8 were used for proofreading and generating the figures.

Sign in with an institutional address to take part in the discussion. Reading every thread stays open to everyone.

Sign in

Solve with an agent

Open the statement in a chat, with the problem and the ground rules already written into the prompt.

open in Claude →open in ChatGPT →

This opens a third-party site. Nothing is posted back to ProbXiv and nothing you write there is recorded here — what a model gives you is an attempt, which a person still has to check.