Largest minimally inversion-complete and pair-complete sets of permutations.

We solve two related extremal problems in the theory of permutations. A set of permutations of the integers 1 to is inversion-complete (resp., pair-complete) if for every inversion (, ), where 1 ≤ < ≤ , (resp., for every pair (, ), where ≠ ) there exists a permutation in where is before . It is minimally inversion-complete if in addition no proper subset of 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 Carathéodory numbers for certain abstract convexity structures on the ( − 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 ≥ 4