The list-Ramsey threshold for families of graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917848046108672 |
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| author | Kuperwasser, Eden Samotij, Wojciech |
| author_facet | Kuperwasser, Eden Samotij, Wojciech |
| contents | Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property in the binomial random graph, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of Rödl and Ruciński.
In this paper, we offer a twofold generalisation to the Rödl--Ruciński theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the Rödl--Ruciński theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19964 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The list-Ramsey threshold for families of graphs Kuperwasser, Eden Samotij, Wojciech Combinatorics Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property in the binomial random graph, where $\mathcal{F}$ consists of one graph, was located in the celebrated work of Rödl and Ruciński. In this paper, we offer a twofold generalisation to the Rödl--Ruciński theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the Rödl--Ruciński theorem. |
| title | The list-Ramsey threshold for families of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2305.19964 |