Main metric invariants of finite metric spaces. III (Q2404916)

Let \({\mathcal K}\) be a family of finite metric spaces of the same cardinality \(N> 1\). Define the functions (for \(X\in {\mathcal K}\)) \textbf{I)} \(\text{Ld}_{mnk}(X)\)=the \(m\)-th element of the increasingly ordered list of \(\text{Ld}_n(S)\), obtained by searching all subsets \(S\subset X\) of the fixed power \(k\), where \(\text{Ld}_n(S)\) is the \(n\)-th element of the list of all distances of different points of \(S\) \textbf{II)} \(\text{Lc}_{lqr}\)=the \(l\)-th element of the increasingly ordered list of \(\text{Lc}_q(S)\), obtained by searching all subsets \(S\subset X\) of the fixed power \(r\), where \(\text{Lc}_q(S)\) is the \(q\)-th element of the list of all distances between points of \(S\) and of \(X\setminus S\) \textbf{III)} \(\text{Lr}_{ijs}\)=the \(i\)-th element of the increasingly ordered list of \(\text{R}_j(S)\), obtained by searching all subsets \(S\subset X\) of the fixed power \(s\), where \(\text{R}_j(S)\) is the \(j\)-radius of \(S\). The main result in this paper is the following Theorem. Let \(2\leq k\leq N\), \(1\leq n\leq C_k^2\), \(1\leq m\leq C_N^k\), \(1\leq r\leq [N/2]\), \(1\leq q\leq r(N-r)\), \(1\leq l\leq C_N^r\), \(3\leq s\leq N\), \(1\leq j\leq s-1\), \(1\leq i\leq C_N^s\). Then, the functions \(\text{Ld}_{mnk}\), \(\text{Lc}_{lqr}\), \(\text{Lr}_{ijs}\) are main metric invariants over the family \({\mathcal K}\). The obtained result is useful for a detailed classification of finite metric spaces. For parts I and II see [Russ. Math. 59, No. 5, 38--40 (2015; Zbl 1321.54054); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2015, No. 5, 45--48 (2015) and ibid. 60, No. 6, 75--78 (2016; Zbl 1347.54043); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2016, No. 6, 86--90 (2016)].