Proof of the Kohayakawa--Kreuter conjecture for the majority of cases
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915673416925184 |
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| author | Bowtell, Candida Hancock, Robert Hyde, Joseph |
| author_facet | Bowtell, Candida Hancock, Robert Hyde, Joseph |
| contents | For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G_{n,p} \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. We reduce the $0$-statement of the Kohayakawa--Kreuter conjecture to a natural deterministic colouring problem and resolve this problem for almost all cases, which in particular includes (but is not limited to) when $H_2$ is strictly $2$-balanced and either has density greater than $2$ or is not bipartite. In addition, we extend our reduction to hypergraphs, proving the colouring problem in almost all cases there as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16760 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proof of the Kohayakawa--Kreuter conjecture for the majority of cases Bowtell, Candida Hancock, Robert Hyde, Joseph Combinatorics For graphs $G, H_1,\dots,H_r$, write $G \to (H_1, \ldots, H_r)$ to denote the property that whenever we $r$-colour the edges of $G$, there is a monochromatic copy of $H_i$ in colour $i$ for some $i \in \{1,\dots,r\}$. Mousset, Nenadov and Samotij proved an upper bound on the threshold function for the property that $G_{n,p} \to (H_1,\dots,H_r)$, thereby resolving the $1$-statement of the Kohayakawa--Kreuter conjecture. We reduce the $0$-statement of the Kohayakawa--Kreuter conjecture to a natural deterministic colouring problem and resolve this problem for almost all cases, which in particular includes (but is not limited to) when $H_2$ is strictly $2$-balanced and either has density greater than $2$ or is not bipartite. In addition, we extend our reduction to hypergraphs, proving the colouring problem in almost all cases there as well. |
| title | Proof of the Kohayakawa--Kreuter conjecture for the majority of cases |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.16760 |