Hypergraphs of Low Degree Without Independent Transversals

For every fixed $k\ge 3$ and every integer $b\ge 2$, we construct a finite vertex-partitioned $k$-uniform hypergraph with blocks of size $b$ and no independent transversal. We show that the maximum number of edges meeting a block is $\left(\frac{(k-1)^{k-1}}{k^k}+O_k(b^{-1})\right)b^k,$ while the ma