---

Claim. Theorem 5 of the paper (printed p. 181) states that for an arbitrary ordinal $\xi$, every $k<\omega$ and every $t$ with $1\le t<\omega$,

$$\omega_{\xi+1}^{(t+1)(k+1)}\to(\mu,t+2)^2 \quad\text{for every }\mu<\omega_{\xi+1}^{k+2}.$$

Unlike Theorems 1 and 4 of the paper, which open with the generalized continuum hypothesis as an assumption, Theorem 5 carries no hypothesis beyond ZFC. With $\xi=0$, $k=0$, $t=1$ and $\mu=\omega_1\omega<\omega_1^2$ it gives

$$\omega_1^2\to(\omega_1\omega,3)^2 ,$$

that is, every graph on the ordinal $\omega_1^2$ has an independent set of order type $\omega_1\omega$ or contains a triangle. This is the relation the site's commentary credits to Erdős and Hajnal, and the one Erdős records under Problem 3 of Erdős 1987 (printed p. 223) as proved by Hajnal and himself, adding that they could never show $\omega_1^2\to(\omega_1\omega,4)^2$.

Covers. The second question of Problem 597, the relation for finite $G$, for $G=K_3$ and hence for every graph on at most three vertices, each being a subgraph of $K_3$. It says nothing about the first question, the infinite targets on at most $\aleph_1$ vertices, and nothing about finite $G$ on more than three vertices, where the ZFC question is open from $K_4-e$ and $C_4$ on.

Source. P. Erdős and A. Hajnal, Ordinary partition relations for ordinal numbers, Periodica Mathematica Hungarica 1 (1971), no. 3, 171–185, doi:10.1007/BF02029142; the issue is dated September 1971 and carries no day, so this page is dated the first of that month. The second paper link is the open scan in the Rényi Institute's Erdős archive. The paper proves Theorem 5 by induction on $t$; nothing on this page is independently reviewed by this project.

Acceptance. Refereed: a journal paper in Periodica Mathematica Hungarica. The site labels Problem 597 OPEN, so the curator's commentary crediting the relation is not acceptance of a claim and reviewed is not listed. White's report uses the relation as the base case of its block-graph theorem.