Abstract:We develop the framework of $\alpha$-largeness introduced by Ketonen and Solovay, by proving a partition theorem for $\alpha$-large sets with $\alpha < \epsilon_0$ which generalizes theorems from Ketonen and Solovay and from Bigorajska and Kotlarski. We also prove that for every $\omega^{nk+3}$-large set $X$ with $\min X \geq 18$, every coloring $f : [X]^2 \to k$ admits an $\omega^n$-large $f$-homogeneous subset. This bound is tight, up to an additive constant.
From: Ludovic Patey [view email]
[v1]
Mon, 9 Feb 2026 15:19:42 UTC (30 KB)
[v2]
Mon, 24 Aug 2026 06:48:03 UTC (32 KB)
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