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Statement

Problem 1 (printed p. 223) asks to determine the ordinals $\alpha$ for which

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

that is, for which every red/blue coloring of the pairs from $\omega^\alpha$ has a red set of order type $\omega^\alpha$ or a blue triangle. The print adds that "$\alpha$ must be a power of $\omega$", offers a prize for a complete characterization and another for $\alpha=\omega^2$, which it calls "the first open case", and cites J. Larson, A short proof of a partition theorem for the ordinal $\omega^\omega$, Ann. Math. Logic 6 (1973/74).

The paper gives no proof or partial result beyond these remarks.

Source. P. Erdős, Some problems on finite and infinite graphs, Logic and Combinatorics (Arcata, Calif., 1985), Contemp. Math. 65, Amer. Math. Soc. (1987), 223--228; Problem 1, p. 223, PDF p. 1 of the Rényi archive's scan (printed p. $n$ = PDF p. $n-222$), read on the rendered page image. The edition read is identified in the source digest.

Read depth. Claims checked: the problem was read clause by clause on the page image. A question has no proof to check.

Proof pointer

None in the source.

Dependencies

None.

Bears on

• Problem 592: the request for a complete characterization is this problem's question, with the print's $\alpha$ in the role of the problem's $\beta$; the site asks for the countable $\beta$, and the print leaves $\alpha$ unqualified. The paper poses it and records no result on it beyond the remark that $\alpha$ must be a power of $\omega$.
• Problem 591: the prize case $\alpha=\omega^2$, the relation $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$, is this problem's question; the paper calls it the first open case and records no result on it.