  There has been interest recently in maximizing the number of independent sets
in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the
number of independent sets in a $d$-regular graph. Similarly, it is a corollary
of the Kruskal-Katona theorem that the lex graph has the maximum number of
independent sets in a graph of fixed size and order. In this paper we solve two
equivalent problems.
  The first is: what $3$-uniform hypergraph on a ground set of size $n$, having
at least $t$ edges, has the most $2$-independent sets? Here a $2$--independent
set is a subset of vertices containing fewer than $2$ vertices from each edge.
This is equivalent to the problem of determining which graph on $n$ vertices
having at least $t$ triangles has the most independent sets. The (hypergraph)
answer is that, ignoring some transient and some persistent exceptions, a
$(2,3,1)$-lex style $3$-graph is optimal.
  We also discuss the problem of maximizing the number of $s$-independent sets
in $r$-uniform hypergraphs of fixed size and order, proving some simple
results, and conjecture an asymptotically correct general solution to the
problem.
