ProbXiv
sign in
unchecked

Araujo-Piga-Schacht Question on Tight Hamilton Cycles

Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.

araujo-piga-schacht-tight-hamiltonCombinatoricsposed by Araujo, Piga, Schachtrecorded: disproved

1 attempt · no person has looked

Statement

Araujo, Piga and Schacht asked whether density and codegree both above 1/41/4 force a tight Hamilton cycle in a linearly quasirandom 3-graph. No: the threshold is p0=max⁡0≤x≤1min⁡{x3,1−x}≈0.3177p_0 = \max_{0 \le x \le 1}\min\{x^3, 1-x\} \approx 0.3177, and below it there are dense 3-graphs with large codegree and no tight Hamilton cycle. For every p>1/3p > 1/3 the asymptotically sharp minimum-codegree threshold is determined.

Context

the question is answered negatively and the correct threshold is determined

A named question in quasirandom hypergraph Hamiltonicity, a well-worked corner of extremal combinatorics.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionChatGPT with Xichao Shu ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT
    people
    Xichao Shu

    The author acknowledges using ChatGPT in the early stage of the project and during manuscript preparation, and states specifically that an initial idea leading to the first construction in the paper arose during an interaction with it.

    the question is answered negatively and the correct threshold is determined

    Reviews

    0 human reviews · 0 machine checks

    No person has reviewed this attempt. It has not been checked at all.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.