Note on polychromatic coloring of hereditary hypergraph families II
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911626313072640 |
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| author | Pálvölgyi, Dömötör |
| author_facet | Pálvölgyi, Dömötör |
| contents | We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy restricted subhypergraphs are $2$-colorable. Together with the previously known case $h=3$, this gives examples with uniformity $2h-1$ for every $h\ge 3$. The construction is based on complements of suitable $h$-uniform hypergraphs on $3h-1$ vertices. For $h\ge 9$ we prove existence by a simple probabilistic argument; the remaining cases $4\le h\le 8$ are certified by a short exhaustive computer check, whose fully reproducible description and source code are included in the appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24412 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Note on polychromatic coloring of hereditary hypergraph families II Pálvölgyi, Dömötör Combinatorics Discrete Mathematics We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy restricted subhypergraphs are $2$-colorable. Together with the previously known case $h=3$, this gives examples with uniformity $2h-1$ for every $h\ge 3$. The construction is based on complements of suitable $h$-uniform hypergraphs on $3h-1$ vertices. For $h\ge 9$ we prove existence by a simple probabilistic argument; the remaining cases $4\le h\le 8$ are certified by a short exhaustive computer check, whose fully reproducible description and source code are included in the appendix. |
| title | Note on polychromatic coloring of hereditary hypergraph families II |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2604.24412 |