On determinants of tournaments and ${\mathcal{D}}_k$

Let $T$ be a tournament with $n$ vertices $v_1,\dots ,v_n$. The skew-adjacency matrix of $T$ is the $n\times n$ zero-diagonal matrix $S_T=[s_{ij}]$ in which $s_{ij}=-s_{ji}=1$ if $v_i$ dominates $v_j$. We define the determinant $\det (T)$ of $T$ as the determinant of $S_T$. It is well-known that $\det (T)=0$ if $n$ is odd and $\det (T)$ is the square of an odd integer if $n$ is even. Let ${\mathcal{D}}_k$ be the set of tournaments whose all subtournaments have determinant at most $k^2$, where $k$ is a positive odd integer. The necessary and sufficient condition for $T\in {\mathcal{D}}_1$ or $T\in {\mathcal{D}}_3$ has been characterized in $2023$. In this paper, we characterize the set ${\mathcal{D}}_5$, obtain some properties of ${\mathcal{D}}_k$. Moreover, for any positive odd integer $k$, we give a construction of a tournament $T$ satisfying that $\det (T)=k^2$, and $T\in {\mathcal{D}}_k\mathrm{\backslash }{\mathcal{D}}_{k-2}$ if $k\ge 3$, which implies ${\mathcal{D}}_k\mathrm{\backslash }{\mathcal{D}}_{k-2}$ is not an empty set for $k\ge 3$.