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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.16599 |
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| _version_ | 1866908720135405568 |
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| author | Yi, Aijun Luo, Zhidan |
| author_facet | Yi, Aijun Luo, Zhidan |
| contents | In 1978, Chung and Liu generalized the definition of the Ramsey number. They introduced the $d$-chromatic Ramsey number as follows. Let $1\leq d< c$ be integers and let $A_{1}, \dots, A_{t}$ be subsets with size $d$ of $[c]$, where $t= {c\choose d}$. For given graphs $G_{1}, \dots, G_{t}$, {\it the $d$-chromatic Ramsey number}, $r^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $N$ such that every $c$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$. The {\it star-critical $d$-chromatic Ramsey number}, $r_{*}^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $k$ such that every $c$-coloring of $E(K_{N}- K_{1, N- 1- k})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$, where $N= r^{d, c}(G_{1}, \dots, G_{t})$. If $G_{1}, \dots, G_{t}= G$, then we simplify them as $r^{d, c}(G)$ (it also call {\it the weakened Ramsey number}) and $r^{d, c}_{*}(G)$, respectively. In this paper, we determine all the value of $r^{d, c}(K_{1, n})$, $r_{*}^{d, c}(K_{1, n})$ and part of the value of $r^{d, c}(K_{1, n_{1}}, \dots, K_{1, n_{t}})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16599 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $d$-chromatic Ramsey number for stars Yi, Aijun Luo, Zhidan Combinatorics In 1978, Chung and Liu generalized the definition of the Ramsey number. They introduced the $d$-chromatic Ramsey number as follows. Let $1\leq d< c$ be integers and let $A_{1}, \dots, A_{t}$ be subsets with size $d$ of $[c]$, where $t= {c\choose d}$. For given graphs $G_{1}, \dots, G_{t}$, {\it the $d$-chromatic Ramsey number}, $r^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $N$ such that every $c$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$. The {\it star-critical $d$-chromatic Ramsey number}, $r_{*}^{d, c}(G_{1}, \dots, G_{t})$, is the minimum positive integer $k$ such that every $c$-coloring of $E(K_{N}- K_{1, N- 1- k})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [t]$, where $N= r^{d, c}(G_{1}, \dots, G_{t})$. If $G_{1}, \dots, G_{t}= G$, then we simplify them as $r^{d, c}(G)$ (it also call {\it the weakened Ramsey number}) and $r^{d, c}_{*}(G)$, respectively. In this paper, we determine all the value of $r^{d, c}(K_{1, n})$, $r_{*}^{d, c}(K_{1, n})$ and part of the value of $r^{d, c}(K_{1, n_{1}}, \dots, K_{1, n_{t}})$. |
| title | The $d$-chromatic Ramsey number for stars |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.16599 |