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We solve two related extremal problems in the theory of permutations. A set Q of per-\udmutations of the integers 1 to n is inversion-complete (resp., pair-complete) if for every\udinversion (j; i), where 1 <= i < j <= n (resp., for every pair (i<> j), where i 6= j), there exists\uda permutation in Q where j is before i. It is minimally inversion-complete if in addi-\udtion no proper subset of Q is inversion-complete; and similarly for pair-completeness. The\udproblems we consider are to determine the maximum cardinality of a minimal inversion-\udcomplete set of permutations, and that of a minimal pair-complete set of permutations.\udThe latter problem arises in the determination of the Caratheodory numbers for certain\udabstract convexity structures on the (n1)-dimensional real and integer vector spaces.\udUsing Mantel's Theorem on the maximum number of edges in a triangle-free graph, we\uddetermine these two maximum cardinalities and we present a complete description of the\udoptimal sets of permutations for each problem. Perhaps surprisingly (since there are twice\udas many pairs to cover as inversions), these two maximum cardinalities coincide when-\udever n>=4

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations