On determinants of tournaments and Dₖ

Let T be a tournament with n vertices v₁,…,vₙ. The skew-adjacency matrix of T is the n× n zero-diagonal matrix ST = [sᵢⱼ] in which sᵢⱼ=-sⱼᵢ=1 if vᵢ dominates vⱼ. We define the determinant (T) of T as the determinant of ST. It is well-known that (T)=0 if n is odd and (T) is the square of an odd integer if n is even. Let Dₖ be the set of tournaments whose all subtournaments have determinant at most k², where k is a positive odd integer. The necessary and sufficient condition for T∈ D₁ or T∈ D₃ has been characterized in $2023$. In this paper, we characterize the set D₅, obtain some properties of Dₖ. Moreover, for any positive odd integer k, we give a construction of a tournament T satisfying that (T)=k², and T∈ Dₖₖ₋₂ if k≥ 3, which implies Dₖₖ₋₂ is not an empty set for k≥ 3.