Maximizing $2$-Independents Sets in $3$-Uniform Hypergraphs

In this paper we solve three equivalent problems. The first is: what $3$-uniform hypergraph on a ground set of size $n$, having at least $e$ edges, has the most $2$-independent sets? Here is subset vertices containing fewer than $2$ from each edge. This to problem determining for which $\partial^+(\partial_2(\mathcal{H}))$ minimized. $\partial_2({\cdot})$ down-shadow level $2$, and $\partial^+({\cdot})$ denotes up-shadow all levels. in turn graph $n$ triangles independent sets. (hypergraph) answer that, ignoring some transient persistent exceptions can classify completely, $(2,3,1)$-lex style $3$-graph optimal.
 We also discuss general maximizing number $s$-independent sets $r$-uniform hypergraphs fixed order, proving simple results, conjecture an asymptotically correct solution problem.