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=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≥4