Coloring $F$-Free Uniform Hypergraphs

Definitions and mathematical statement

To make the supplied question precise under conventional interpretations, let $k$ be an integer with $k\ge 2$. A finite $k$-uniform hypergraph is a pair $H=(V(H),E(H))$, where $E(H)\subseteq\binom{V(H)}{k}$. Its maximum vertex degree is \[ \Delta(H)=\max_{v\in V(H)}\left|\{e\in E(H):v\in e\}\right|, \] with $\Delta(H)=0$ if $V(H)=\varnothing$. Its weak chromatic number $\chi(H)$ is the least positive integer $q$ for which there is a map $\varphi:V(H)\to\{1,\ldots,q\}$ such that no edge is monochromatic. For a fixed $k$-uniform hypergraph $F$, say that $H$ is $F$-free if there is no injective map $\iota:V(F)\to V(H)$ satisfying $\iota(e)\in E(H)$ for every $e\in E(F)$. Thus this interpretation excludes ordinary subhypergraph copies, without requiring them to be induced.

The supplied question, on the range where its displayed expression is defined, asks whether \[ \begin{gathered} \text{for every integer }k\ge2\text{ and every fixed finite }k\text{-uniform hypergraph }F,\\ \text{there exists }c_F>0\text{ such that every finite }F\text{-free }k\text{-uniform hypergraph }H\\ \text{with }\Delta(H)\ge2\text{ satisfies}\qquad \chi(H)\le c_F\left(\frac{\Delta(H)}{\log\Delta(H)}\right)^{1/(k-1)}. \end{gathered} \] Here $\log$ denotes the natural logarithm, and $c_F$ may depend only on $F$, whose uniformity determines $k$.

Unresolved qualifications. The source does not explicitly specify finiteness, the range of $k$, the copy convention, or the coloring convention; the definitions above describe a conventional reading requiring review. The supplied wording quantifies over every maximum degree, but its expression is undefined at $\Delta=0$ and $\Delta=1$. The displayed formulation therefore records only its meaningful range $\Delta\ge2$; a small-degree convention or an explicitly asymptotic formulation remains to be supplied. No structural restriction on $F$ beyond uniformity is stated, and none has been inserted.

Short English statement

The supplied question asks whether forbidding any fixed $k$-uniform hypergraph $F$ forces the weak chromatic number to be at most a constant depending only on $F$ times $(\Delta/\log\Delta)^{1/(k-1)}$. The draft spells out conventional finite, non-induced-copy interpretations; the intended conventions and treatment of degrees zero and one require review. The source labels the question open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001245$, title "Coloring $F$-Free Uniform Hypergraphs," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2009$ but names no proposer.
• Supplied report citation: Combinatorics and Probability (2009), Oberwolfach Reports, DOI 10.4171/owr/2009/23, pp. 1225--1302. https://doi.org/10.4171/owr/2009/23. The background describes these reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or named contributor is supplied.
• Supplied literature reference: Bohman--Frieze--Mubayi, Coloring simple hypergraphs. https://doi.org/10.1016/j.jctb.2013.09.003.
• Supplied literature reference, labeled "Frieze--Mubayi restricted theorem." https://arxiv.org/abs/0809.2979.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the conjectured bound is proved for important restricted classes, including "simple hypergraphs," while the stated general $F$-free version remains conjectural. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and describes its preparation as meaning-preserving semantic, typographic, and LaTeX cleanup with no unsupported source-dependent content added. These are supplied provenance claims, not mathematical verification of this draft.