  We solve two related extremal problems in the theory of permutations. A set
$Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp.,
pair-complete) if for every inversion $(j,i)$, where $1 \le i \textless{} j \le
n$, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a
permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete
if in addition no proper subset of~$Q$ is inversion-complete; and similarly for
pair-completeness. The problems we consider are to determine the maximum
cardinality of a minimal inversion-complete set of permutations, and that of a
minimal pair-complete set of permutations. The latter problem arises in the
determination of the Carath\'eodory numbers for certain abstract convexity
structures on the $(n-1)$-dimensional real and integer vector spaces. Using
Mantel's Theorem on the maximum number of edges in a triangle-free graph, we
determine these two maximum cardinalities and we present a complete description
of the optimal sets of permutations for each problem. Perhaps surprisingly
(since there are twice as many pairs to cover as inversions), these two maximum
cardinalities coincide whenever $n \ge 4$.
