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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | https://arxiv.org/abs/2312.04955 |
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| _version_ | 1866909229789478912 |
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| author | Boyadzhiyska, Simona Lo, Allan |
| author_facet | Boyadzhiyska, Simona Lo, Allan |
| contents | Given a pair of $k$-uniform hypergraphs $(G,H)$, the Ramsey number of $(G,H)$, denoted by $R(G,H)$, is the smallest integer $n$ such that in every red/blue-colouring of the edges of $K_n^{(k)}$ there exists a red copy of $G$ or a blue copy of $H$. Burr showed that, for any pair of graphs $(G,H)$, where $G$ is large and connected, $R(G,H) \geq (v(G)-1)(χ(H)-1)+σ(H)$, where $σ(H)$ stands for the minimum size of a colour class over all proper $χ(H)$-colourings of $H$. We say that $G$ is $H$-good if $R(G,H)$ is equal to the general lower bound. Burr showed that, for any graph~$H$, every sufficiently long path is $H$-good.
Our goal is to explore the notion of Ramsey goodness in the setting of $k$-uniform hypergraphs. We demonstrate that, in stark contrast to the graph case, $k$-uniform $\ell$-paths are not $H$-good for a large class of $k$-graphs. On the other hand, we prove that long loose paths are always at least asymptotically $H$-good for every $H$ and derive lower and upper bounds that are best possible in a certain sense.
In the 3-uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3-graph $H$ when $H$ belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3-graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04955 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ramsey goodness of $k$-uniform paths, or the lack thereof Boyadzhiyska, Simona Lo, Allan Combinatorics Given a pair of $k$-uniform hypergraphs $(G,H)$, the Ramsey number of $(G,H)$, denoted by $R(G,H)$, is the smallest integer $n$ such that in every red/blue-colouring of the edges of $K_n^{(k)}$ there exists a red copy of $G$ or a blue copy of $H$. Burr showed that, for any pair of graphs $(G,H)$, where $G$ is large and connected, $R(G,H) \geq (v(G)-1)(χ(H)-1)+σ(H)$, where $σ(H)$ stands for the minimum size of a colour class over all proper $χ(H)$-colourings of $H$. We say that $G$ is $H$-good if $R(G,H)$ is equal to the general lower bound. Burr showed that, for any graph~$H$, every sufficiently long path is $H$-good. Our goal is to explore the notion of Ramsey goodness in the setting of $k$-uniform hypergraphs. We demonstrate that, in stark contrast to the graph case, $k$-uniform $\ell$-paths are not $H$-good for a large class of $k$-graphs. On the other hand, we prove that long loose paths are always at least asymptotically $H$-good for every $H$ and derive lower and upper bounds that are best possible in a certain sense. In the 3-uniform setting, we complement our negative result with a positive one, in which we determine the Ramsey number asymptotically for pairs containing a long tight path and a 3-graph $H$ when $H$ belongs to a certain family of hypergraphs. This extends a result of Balogh, Clemen, Skokan, and Wagner for the Fano plane asymptotically to a much larger family of 3-graphs. |
| title | Ramsey goodness of $k$-uniform paths, or the lack thereof |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2312.04955 |