For every $n$ and $0<\delta <1$, we construct graphs on $n$ nodes such that every two sets of size $n^{\delta}$ share an edge, having essentially optimal maximum degree $n^{1-\delta+o(1)}$. Using known and new reductions from these graphs, we derive new explicit constructions of: 1. A $k$ round sorting algorithm using $n^{1+1/k+o(1)}$ comparisons. 2. A $k… (More)