Erdős Problem #118

**Erdős Problem 118** (problem of Erdős and Hajnal). Let \(\alpha\) be such that \(\alpha \to (\alpha, 3)^2\). For every \(n \ge 3\), must \(\alpha \to (\alpha, n)^2\)? The original question is asked for cardinals, ordinals, and order types; we state the ordinal version, which is the standard reading: ordinals \(\alpha\) with \(\alpha \to (\alpha, 3)^2\) are called *partition ordinals*. The answer is **no**, shown independently by Schipperus [Sc99, Sc10] and Darby [Da99]. Larson [La00] gave the concrete counterexample \(\alpha = \omega^{\omega^2}\), for which \(\alpha \to (\alpha, 3)^2\) holds yet \(\alpha \not\to (\alpha, 5)^2\). See Chapter 2.9 of [HST10] for further background.