$K_4^-$-free triple systems without large stars in the complement
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913783404822528 |
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| author | Mubayi, Dhruv Spanier, Nicholas |
| author_facet | Mubayi, Dhruv Spanier, Nicholas |
| contents | The $n$-star $S_n$ is the $n$-vertex triple system with ${n-1 \choose 2}$ edges all of which contain a fixed vertex, and $K_4^-$ is the unique triple system with four vertices and three edges. We prove that the Ramsey number $r(K_4^-, S_n)$ has order of magnitude $n^2 /\log n$.
This confirms a conjecture of Conlon, Fox, He, Suk, Verstraëte and the first author. It also generalizes the well-known bound of Kim for the graph Ramsey number $r(3,n)$, as the link of any vertex in a $K_4^-$-free triple system is a triangle-free graph. Our method builds on the approach of Guo and Warnke who adapted Kim's lower bound for $r(3,n)$ to the pseudorandom setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06076 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $K_4^-$-free triple systems without large stars in the complement Mubayi, Dhruv Spanier, Nicholas Combinatorics 05C55, 05C65, 05D10, 05D40 The $n$-star $S_n$ is the $n$-vertex triple system with ${n-1 \choose 2}$ edges all of which contain a fixed vertex, and $K_4^-$ is the unique triple system with four vertices and three edges. We prove that the Ramsey number $r(K_4^-, S_n)$ has order of magnitude $n^2 /\log n$. This confirms a conjecture of Conlon, Fox, He, Suk, Verstraëte and the first author. It also generalizes the well-known bound of Kim for the graph Ramsey number $r(3,n)$, as the link of any vertex in a $K_4^-$-free triple system is a triangle-free graph. Our method builds on the approach of Guo and Warnke who adapted Kim's lower bound for $r(3,n)$ to the pseudorandom setting. |
| title | $K_4^-$-free triple systems without large stars in the complement |
| topic | Combinatorics 05C55, 05C65, 05D10, 05D40 |
| url | https://arxiv.org/abs/2504.06076 |