Optimal fractional chromatic bound for degenerate linear hypergraphs
Determine whether, for every integer k ≥ 3 and every ε > 0, there exists d₀ such that every k-uniform d-degenerate linear hypergraph satisfies χ_f(𝓗) ≤ (1 + ε)((k − 1)d/log d)^{1/(k−1)} for all d ≥ d₀.
References
It is natural to ask whether the bound in Theorem~\ref{thm: degenerate}\ref{item: degenerate ub} extends to all linear hypergraphs; that is, whether we can remove the restriction on cycle configurations.
\begin{problem}\label{problem: degeneracy linear} Let $k \ge 3$ be an integer, and let $\eps > 0$ be arbitrary. Does there exist $d_0 \in \N$ such that the following holds for every $d \ge d_0$? If $\cH$ is a $k$-uniform $d$-degenerate linear hypergraph, then
\chi_f(\cH) \le (1+\eps)\left((k-1) \, \frac{d}{\log d}\right){\frac{1}{k-1}.
\end{problem}