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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.11574 |
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| _version_ | 1866908055155769344 |
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| author | Araujo, Igor Peng, Dadong |
| author_facet | Araujo, Igor Peng, Dadong |
| contents | Erdős and Rado [P. Erdős, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249-255] introduced the Canonical Ramsey numbers $\text{er}(t)$ as the minimum number $n$ such that every edge-coloring of the ordered complete graph $K_n$ contains either a monochromatic, rainbow, upper lexical, or lower lexical clique of order $t$. Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial Theory Series B 80 (2000) 172-177] introduced the unordered asymmetric version of the Canonical Ramsey numbers $\text{CR}(s,r)$ as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a rainbow clique of order $r$, or an orderable clique of order $s$.
We show that $\text{CR}(s,r) = O(r^3/\log r)^{s-2}$, which, up to the multiplicative constant, matches the known lower bound and improves the previously best known bound $\text{CR}(s,r) = O(r^3/\log r)^{s-1}$ by Jiang [T. Jiang, Canonical Ramsey numbers and proporly colored cycles, Discrete Mathematics 309 (2009) 4247-4252]. We also obtain bounds on the further variant $\text{ER}(m,\ell,r)$, defined as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a monochromatic $K_m$, lexical $K_\ell$, or rainbow $K_r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11574 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the off-diagonal unordered Erdős-Rado numbers Araujo, Igor Peng, Dadong Combinatorics Erdős and Rado [P. Erdős, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249-255] introduced the Canonical Ramsey numbers $\text{er}(t)$ as the minimum number $n$ such that every edge-coloring of the ordered complete graph $K_n$ contains either a monochromatic, rainbow, upper lexical, or lower lexical clique of order $t$. Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial Theory Series B 80 (2000) 172-177] introduced the unordered asymmetric version of the Canonical Ramsey numbers $\text{CR}(s,r)$ as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a rainbow clique of order $r$, or an orderable clique of order $s$. We show that $\text{CR}(s,r) = O(r^3/\log r)^{s-2}$, which, up to the multiplicative constant, matches the known lower bound and improves the previously best known bound $\text{CR}(s,r) = O(r^3/\log r)^{s-1}$ by Jiang [T. Jiang, Canonical Ramsey numbers and proporly colored cycles, Discrete Mathematics 309 (2009) 4247-4252]. We also obtain bounds on the further variant $\text{ER}(m,\ell,r)$, defined as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a monochromatic $K_m$, lexical $K_\ell$, or rainbow $K_r$. |
| title | On the off-diagonal unordered Erdős-Rado numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.11574 |