On the Ramsey numbers of fans and stars
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918414281342976 |
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| author | DeBiasio, Louis Wimbish, Tucker |
| author_facet | DeBiasio, Louis Wimbish, Tucker |
| contents | Let $F_n$ be the graph on $2n+1$ vertices consisting of $n$ triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of $F_n$ is between $4.5n-5$ (Chen, Yu, Zhao) and $(5+\frac{1}{6})n+O(1)$ (Dvo{ř}{á}k and Metrebian). We improve both of these bounds as follows $$4.732n\approx (3+\sqrt{3})n-8< R(F_n)\leq (5+o(1))n.$$
Additionally, as it relates to the lower bound on $R(F_n)$ (and for which nothing was known when $n< m< n(n-1)$), we determine the Ramsey numbers of stars vs.~fans, within a constant, as follows $$R(K_{1,m}, F_{n})=
\begin{cases}
m+2n-\frac{1+(-1)^{m}}{2}, & m\leq n
\frac{3m+\sqrt{m^2+8n^2}}{2}+Θ(1), & m>n
\end{cases}. $$ In particular, we have $R(K_{1,2n}, F_n)=(3+\sqrt{3})n+Θ(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27110 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Ramsey numbers of fans and stars DeBiasio, Louis Wimbish, Tucker Combinatorics Let $F_n$ be the graph on $2n+1$ vertices consisting of $n$ triangles meeting at a single vertex. After a number of improvements over the years, it is currently known that the Ramsey number of $F_n$ is between $4.5n-5$ (Chen, Yu, Zhao) and $(5+\frac{1}{6})n+O(1)$ (Dvo{ř}{á}k and Metrebian). We improve both of these bounds as follows $$4.732n\approx (3+\sqrt{3})n-8< R(F_n)\leq (5+o(1))n.$$ Additionally, as it relates to the lower bound on $R(F_n)$ (and for which nothing was known when $n< m< n(n-1)$), we determine the Ramsey numbers of stars vs.~fans, within a constant, as follows $$R(K_{1,m}, F_{n})= \begin{cases} m+2n-\frac{1+(-1)^{m}}{2}, & m\leq n \frac{3m+\sqrt{m^2+8n^2}}{2}+Θ(1), & m>n \end{cases}. $$ In particular, we have $R(K_{1,2n}, F_n)=(3+\sqrt{3})n+Θ(1)$. |
| title | On the Ramsey numbers of fans and stars |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.27110 |