Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space \(\mathbb {R}^{p,q}\) ${\mathbb{R}}^{p,q}$ . We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value \(\begin{aligned} || \varvec{x}-\varvec{y}||&=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \\&\quad -(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{aligned}\) $$\begin{array}{cc}\hfill ||\mathit{x}-\mathit{y}||& ={(x_1-y_1)}^2+\cdots +{(x_p-y_p)}^2\hfill \\ \hfill & -{(x_{p+1}-y_{p+1})}^2-\cdots -{(x_{p+q}-y_{p+q})}^2\hfill \end{array}$$ instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions.