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Claim. There is $c>0$ such that every $n$-vertex graph with no induced five-vertex path $P_5$ has a clique or a stable set of size at least $n^c$. This is Theorem 1.2 of T. Nguyen, A. Scott and P. Seymour, Induced subgraph density. VII. The five-vertex path, Proc. Lond. Math. Soc. (3) 132 (2026), no. 3, e70133, first posted as arXiv:2312.15333 on 2023-12-23 (the claim's date), proved in the stronger form of its Theorem 1.5, that $P_5$ has the polynomial Rödl property; the corpus's card records both statements. It is the question of Problem 61 for $H=P_5$ and, since a graph is $P_5$-free exactly when its complement has no induced copy of the complement of $P_5$ (the house), for $H$ the house. The paper's introduction (p. 1) draws the consequence that this page records as its second part: by Alon, Pach and Solymosi's substitution theorem the five-vertex case reduces to the prime five-vertex graphs, the bull, $C_5$ and $P_5$ with its complement; Erdős and Hajnal had every graph on at most four vertices, Chudnovsky and Safra the bull and Chudnovsky, Scott, Seymour and Spirkl $C_5$, so the conjecture holds for every graph on at most five vertices.

Covers. $H=P_5$ and $H$ the house and, with the four pages under Depends on., every $H$ on at most five vertices. The problem stays open for larger $H$; the two six-vertex cases claimed later by Huang, Ju and Zhou have their own claim page, linked from the problem page.

Depends on. For the five-vertex conclusion only: Erdős and Hajnal's cases on at most four vertices, Alon, Pach and Solymosi's substitution closure, Chudnovsky and Safra's bull-free case and Chudnovsky, Scott, Seymour and Spirkl's five-cycle case. The $P_5$ theorem itself rests on no page of this wiki.

Acceptance. The paper is a refereed publication in the Proceedings of the London Mathematical Society, published online 2026-03-23, which is the refereed evidence. The site's commentary credits the case to the paper and draws the same five-vertex conclusion, but the site labels the problem OPEN, so no reviewed evidence is listed. This corpus has not checked the proof.