Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917461962522624 |
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| author | Mao, Yaping |
| author_facet | Mao, Yaping |
| contents | For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erdős, Faudree, Rousseau, and Schelp: Ramsey-finiteness is preserved by adjoining disjoint matchings, and $(G,H)$ is Ramsey-infinite unless both graphs are odd stars or one graph has a $K_2$ component. We also replace Burr's stronger 1979 survey characterization by the correct necessary-and-sufficient form: apart from the matching case and the odd-star-with-matchings case, the only additional finite pairs are Faudree's star-forest family. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17356 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures Mao, Yaping Combinatorics For finite graphs $G$ and $H$, let $\RR(G,H)$ denote the isomorphism classes of Ramsey-minimal graphs for $(G,H)$. We prove two 1981 conjectures of Burr, Erdős, Faudree, Rousseau, and Schelp: Ramsey-finiteness is preserved by adjoining disjoint matchings, and $(G,H)$ is Ramsey-infinite unless both graphs are odd stars or one graph has a $K_2$ component. We also replace Burr's stronger 1979 survey characterization by the correct necessary-and-sufficient form: apart from the matching case and the odd-star-with-matchings case, the only additional finite pairs are Faudree's star-forest family. |
| title | Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.17356 |