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Record Lower Bounds for the Shannon Capacity of Odd Cycles

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shannon-capacity-odd-cycles-recordsCombinatoricsposed by Claude Shannon, 1956recorded: partial

1 attempt · 1 machine check · no person has looked

Statement

Determine the Shannon capacities of odd cycles beyond C5C_5, or improve the best explicit bounds. Lovasz's theta function settled C5C_5 in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model assistance and formally verified, are Θ(C7)≥3.258805369885\Theta(C_7) \ge 3.258805369885, Θ(C11)≥5.294502522149\Theta(C_{11}) \ge 5.294502522149, Θ(C13)≥6.302455083464\Theta(C_{13}) \ge 6.302455083464, Θ(C15)≥7.301600534487\Theta(C_{15}) \ge 7.301600534487, Θ(C19)≥9.357192705918\Theta(C_{19}) \ge 9.357192705918, Θ(C21)≥10.342455853338\Theta(C_{21}) \ge 10.342455853338 and Θ(C23)≥11.328224257774\Theta(C_{23}) \ge 11.328224257774.

Context

record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5

Shannon capacity of odd cycles, the classic post-Lovász-theta question since 1956.

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  • #1

    Attempt 1

    constructionChatGPT 5.6 Sol Pro, ChatGPT 5.6 Sol, Claude Opus 5 with Pjotr Buys, Sven Polak, Jeroen Zuiddam, Yu Gao, Nathaniel Itty, Christopher D. Rosin, Chase Carstensen, Daniel Reichman ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    ChatGPT 5.6 Sol Pro, ChatGPT 5.6 Sol, Claude Opus 5
    people
    Pjotr Buys, Sven Polak, Jeroen Zuiddam, Yu Gao, Nathaniel Itty, Christopher D. Rosin, Chase Carstensen, Daniel Reichman

    Three model-assisted papers in eleven days, each beating the last. Itty, Rosin, Carstensen and Reichman had ChatGPT-5.6 Sol Pro generate and run search programs across repeated prompts, returning explicit independent sets in strong graph powers that the authors checked. Gao then improved C7C_7 with a recursive construction and states that ChatGPT 5.6 Sol implemented all the code and expanded the proofs. Buys, Polak and Zuiddam followed both methods using ChatGPT 5.6 Sol Pro and Claude Opus 5, beat every previous bound, added three more cycles, and formalised the lot in Lean.

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    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      scope Lean formalization of the result

      The current records are formalised in Lean 4 at the linked repository, one base tuple per bound, so the seven stated inequalities are machine-checked rather than author-checked. We have not compiled it. Gao's intermediate record ships exact-integer verification code pinned to a fixed commit, and the earlier Itty-Rosin-Carstensen-Reichman constructions came with public data, prompts and checking code. All three are arXiv preprints; none is peer-reviewed.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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