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Erdős–Hajnal for the five-vertex path

lax-57·formalized by Édouard Bonnet @EdouardBonnet·created 2026-08-20·GitHub @49cc081·Lean v4.30.0 epoch · mathlib c5ea00351c28

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    Abstract

    This submission formalizes Nguyen, Scott, and Seymour's proof that the five-vertex path P5P_5 has the Erdős–Hajnal property. It proves that there is a positive integer qq such that every finite graph GG with no induced copy of P5P_5 satisfies

    ∣V(G)∣≤max⁡{α(G),ω(G)}q.|V(G)| \leq \max\{\alpha(G),\omega(G)\}^{q}.

    The formalization follows the paper's blockade argument through polynomial semisparse blockades in house-free graphs, the sparse-house trichotomy and its iteration, and a final critical-graph argument. Density inequalities are stated over the natural numbers with denominators cleared. The development uses Rödl's theorem, sparse thinning, maximum-degree reduction, and the bipartite comb lemma from Lax 54; every argument specific to P5P_5 and its complement is proved in this submission.

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    Cite this

    @misc{lax-57,
  author = {Édouard Bonnet},
  title = {Erdős–Hajnal for the five-vertex path},
  year = {2026},
  howpublished = {Lax Archive, lax-57},
  url = {https://laxarchive.org/lax-57/},
  note = {draft},
}

    References

    1. Tung Nguyen, Alex Scott and Paul Seymour. Induced subgraph density. VII. The five-vertex path. 2026. arXiv:2312.15333
    2. Maria Chudnovsky, Alex Scott, Paul Seymour and Sophie Spirkl. Erdős–Hajnal for graphs with no 5-hole. Proceedings of the London Mathematical Society 126(3):997–1014, 2023. doi:10.1112/plms.12504

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