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 Rp,q. We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value ||x − y|| = (x1 − y1)² +···+ (x p − yp)² − (xp+1 − yp+1)² −···− (xp+q − yp+q )² 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.