The independence number of uncrowded hypergraphs: bounds matching the shattering threshold (opens in new tab)

A foundational theorem of Ajtai, Koml\'os, Pintz, Spencer, and Szemer\'edi asserts that every $n$-vertex $k$-uniform uncrowded hypergraph with maximum degree $\Delta$ contains an independent set of size $c_k n{\left(\frac{\log \Delta}{\Delta}\right)^{\frac{1}{k-1}}}$, for some constant $c_k>0$. Determining the optimal leading constant $c_k$ in this bound is a major open problem. A natural target is the so-called shattering-threshold constant $...