Random independent sets in uncrowded hypergraphs

Source abstract

Given any fixed integer $k \ge 2$ and sufficiently large $d$, we show that the largest possible fractional chromatic number of a $k$-uniform $d$-degenerate uncrowded hypergraph $H$ (i.e., with girth at least $5$) satisfies $$χ_f(H) = (1 + o_d(1)) \left((k-1)\,\frac{d}{\log d}\right)^{\frac{1}{k-1}}.$$ In fact, we prove that this holds for $k$-uniform $d$-degenerate hypergraphs of girth at least $g$, for any given $g \ge 5$. As a corollary, we obtain improved bounds on the fractional chromatic number of $d$-degenerate linear hypergraphs. This work builds upon a recent result by Allen, Dhawan, and Noel, extending it from graphs to hypergraphs. In addition to overcoming the new difficulties that arise in the hypergraph setting, our approach yields a simpler proof even in the original graph case. Our proof of the upper bound uses a simpler iterative procedure for sampling independent sets. We also establish bounds for fractional colorings with local demands, a framework introduced by Kelly and Postle, verifying a recent conjecture of Yu and Zhang. As a consequence, we obtain a degree-sequence bound on the independence number of uncrowded hypergraphs with a leading constant matching the shattering threshold. For the matching lower bound, we use a hypergraph variant of the uniform attachment model and harmonic vertex weights to bound the fractional chromatic number via linear programming duality, then remove all short cycles by deleting vertices of negligible total weight. We also replace the Catalan-number argument used in the graph case with a matrix-norm estimate, simplifying the analysis.