On diversity and stability of unit bases for the Euclidean metric

Abstract

We prove the existence of a family Ω(n) of 2 c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E n , 1) of the Euclidean space E n , n ≥ 2, such that (a) for each graph G ϵ Ω(n), any homomorphism of G to (E n , 1) is an isometry of E n ; moreover, for each subgraph G 0 of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G 0 to (E n , 1) is an isometry (of the set of the vertices of G 0); (b) each graph G ϵ Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.