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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.12358 |
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| _version_ | 1866908906507206656 |
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| author | Kucheriya, Gaurav Lo, Allan Petr, Jan Sgueglia, Amedeo Yan, Jun |
| author_facet | Kucheriya, Gaurav Lo, Allan Petr, Jan Sgueglia, Amedeo Yan, Jun |
| contents | An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the ordered Ramsey number $R_<(H)$ is the smallest integer $N$ such that every red/blue edge-colouring of $K_N^<$ contains a monochromatic copy of $H$ with vertices appearing in the same relative order as in $H$. Balko, Cibulka, Král, and Kynčl asked whether, among all ordered paths on $n$ vertices, the ordered Ramsey number is minimised by the alternating path $\mathrm{AP}_n$ -- the ordered path with vertex set $[n]$ such that the vertices encountered along the path are $1, n, 2, n - 1,3, n-2,\dots$. Motivated by this problem, we make progress on establishing the value of $R_<(\mathrm{AP}_n)$ by proving that \[ R_{<}(\mathrm{AP}_n)\leq \left(2+\frac{\sqrt{2}}{2}+o(1)\right)n. \] We then use similar methods to determine the exact ordered Turán number of $\mathrm{AP}_n$, and study the ordered Ramsey and Turán numbers of several related ordered paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12358 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ordered Ramsey and Turán numbers of alternating paths and their variants Kucheriya, Gaurav Lo, Allan Petr, Jan Sgueglia, Amedeo Yan, Jun Combinatorics 05D10 An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the ordered Ramsey number $R_<(H)$ is the smallest integer $N$ such that every red/blue edge-colouring of $K_N^<$ contains a monochromatic copy of $H$ with vertices appearing in the same relative order as in $H$. Balko, Cibulka, Král, and Kynčl asked whether, among all ordered paths on $n$ vertices, the ordered Ramsey number is minimised by the alternating path $\mathrm{AP}_n$ -- the ordered path with vertex set $[n]$ such that the vertices encountered along the path are $1, n, 2, n - 1,3, n-2,\dots$. Motivated by this problem, we make progress on establishing the value of $R_<(\mathrm{AP}_n)$ by proving that \[ R_{<}(\mathrm{AP}_n)\leq \left(2+\frac{\sqrt{2}}{2}+o(1)\right)n. \] We then use similar methods to determine the exact ordered Turán number of $\mathrm{AP}_n$, and study the ordered Ramsey and Turán numbers of several related ordered paths. |
| title | Ordered Ramsey and Turán numbers of alternating paths and their variants |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2603.12358 |