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arXiv · 2604.23433

On closed Ramsey numbers of small countable ordinals

Necdet Duman·Özge Gönül·Burak Kaya·Jayatra Saxena·Yiğithan Tamer
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Abstract

This paper is a contribution to the investigation of closed partition relations for pairs of countable ordinals. As our main result, we prove that \[ω^4 \cdot (n-2)+1 < R^{cl}(ω\cdot n+1,3)<ω^5\] for every integer $n \geq 3$. This result significantly improves the existing upper and lower bounds for these closed Ramsey numbers. In addition, we prove that \[ω^θ\nrightarrow_{cl} (ω^α,3)^2\] whenever $1 \leq α\leq θ<ω_1$ satisfy $θ< R(α,3)$. This result asymptotically improves the existing lower bounds for $R^{cl}(ω^n,3)$ and slightly strengthens the existing necessary condition for being a topological partition ordinal.

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Necdet Duman, Özge Gönül, Burak Kaya, Jayatra Saxena, Yiğithan Tamer. 2026-04-25. On closed Ramsey numbers of small countable ordinals. https://arxiv.org/abs/2604.23433

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