Verstraëte–Wilson conjecture on independent sets in linear hypergraphs

Let $k\geq 3$, and let $a_k>0$ satisfy $a_k=1-o_k(1)$ as $k\to\infty$. A $k$-uniform hypergraph is linear if it contains no $2$-cycle. Verstraëte–Wilson conjecture. Every $n$-vertex $k$-uniform linear hypergraph ${\mathcal H}$ of maximum degree $\Delta$ satisfies

$$\alpha({\mathcal H})\geq (a_k-o_\Delta(1))\,n\left(\frac{\log\Delta}{\Delta}\right)^{\frac{1}{k-1}}.$$

This conjecture concerns extending shattering-threshold-matching independence bounds from uncrowded to linear hypergraphs; the paper subsequently formulates a stronger asymptotic version and explains that existing random-sampling reductions do not resolve it.