On the largest chromatic number of $F$-free hypergraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917431143825408 |
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| author | Wang, Yichen Duan, Mengyu Gerbner, Dániel Karim, Hilal Hama |
| author_facet | Wang, Yichen Duan, Mengyu Gerbner, Dániel Karim, Hilal Hama |
| contents | Given a hypergraph $F$, what is the largest chromatic number that an $F$-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if $F$ contains a cycle, and the largest chromatic number of $F$-free graphs is $k-1$ if $F$ is a forest on $k$ vertices. The situation is more complicated for hypergraphs.
The strong coloring of a hypergraph is a coloring of the vertices such that every hyperedge is rainbow. The weak coloring of a hypergraph is a coloring of the vertices such that no hyperedge is monochromatic. The strong/weak chromatic number of a hypergraph is the minimum number of colors in a strong/weak coloring of the hypergraph. Our question has been completely answered for the weak chromatic number, similarly to the graph case.
We characterize the hypergraphs $F$ such that $F$-free hypergraphs have bounded strong chromatic number. The only remaining case is when $F$ is the 3-uniform expansion $S_k^+$ of a star with $k$ edges. Concerning the strong chromatic number of $S_k^+$-free hypergraphs, we give bounds that are asymptitically sharp as $k\rightarrow\infty$.
We also consider the same problem when the Berge copies of a graph $F$ are forbidden. We characterize when the strong/weak chromatic numbers are bounded in this case, and obtain sharp results or bounds for specific trees. In particular, when $F$ is a path, we give a tight bound when $r=3$ and an asymptotically sharp bound when $r=4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_21551 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the largest chromatic number of $F$-free hypergraphs Wang, Yichen Duan, Mengyu Gerbner, Dániel Karim, Hilal Hama Combinatorics 05C15, 05C85 Given a hypergraph $F$, what is the largest chromatic number that an $F$-free hypergraph can have? In the case of graphs, this question is easy to answer: the chromatic number is unbounded if $F$ contains a cycle, and the largest chromatic number of $F$-free graphs is $k-1$ if $F$ is a forest on $k$ vertices. The situation is more complicated for hypergraphs. The strong coloring of a hypergraph is a coloring of the vertices such that every hyperedge is rainbow. The weak coloring of a hypergraph is a coloring of the vertices such that no hyperedge is monochromatic. The strong/weak chromatic number of a hypergraph is the minimum number of colors in a strong/weak coloring of the hypergraph. Our question has been completely answered for the weak chromatic number, similarly to the graph case. We characterize the hypergraphs $F$ such that $F$-free hypergraphs have bounded strong chromatic number. The only remaining case is when $F$ is the 3-uniform expansion $S_k^+$ of a star with $k$ edges. Concerning the strong chromatic number of $S_k^+$-free hypergraphs, we give bounds that are asymptitically sharp as $k\rightarrow\infty$. We also consider the same problem when the Berge copies of a graph $F$ are forbidden. We characterize when the strong/weak chromatic numbers are bounded in this case, and obtain sharp results or bounds for specific trees. In particular, when $F$ is a path, we give a tight bound when $r=3$ and an asymptotically sharp bound when $r=4$. |
| title | On the largest chromatic number of $F$-free hypergraphs |
| topic | Combinatorics 05C15, 05C85 |
| url | https://arxiv.org/abs/2604.21551 |