Above and below
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918524688007168 |
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| author | Chen, Wenchong Pohoata, Cosmin |
| author_facet | Chen, Wenchong Pohoata, Cosmin |
| contents | We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{Θ(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27061 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Above and below Chen, Wenchong Pohoata, Cosmin Combinatorics We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{Θ(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko. |
| title | Above and below |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.27061 |