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On the average size of independent sets in triangle-free graphs

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on-the-average-size-of-independent-sets-in-triangle-free-graphs-4Combinatoricsmath.COposed by Ewan Davies, Matthew Jenssen, Will Perkins, Barnaby Robertsrecorded: open

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Statement

For every KrK_{r} -free graph G, α(G)α‾G(1)≥1+1r.\frac{\alpha(G)}{\overline{\alpha}_{G}(1)}\geq 1+\frac{1}{r}.

For every KrK_{r} -free graph G of minimum degree d, α(G)α‾G(1)≥2−od(1),\frac{\alpha(G)}{\overline{\alpha}_{G}(1)}\geq 2-o_{d}(1), with r fixed as d→∞d \to\infty .

Context

Candidate 4 of the open problems stated in "On the average size of independent sets in triangle-free graphs", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationShengtong Zhang ·
    AI involvement
    ai discovered
    — the result was found by a model.
    people
    Shengtong Zhang

    A complete curated solution is available in the attached PDF.

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