Caltech/UCLA Joint Analysis Seminar

UCLA - MS 7608

A distance graph Γ is a graph with vertices in a Euclidean space, with edges made of rigid rods that can freely turn around the vertices. An isometric embedding of Γ into a set A can be visualized as a folding of Γ such that all of its vertices are supported by A. We show that if Γ is k-degenerate, i.e. if all of its sub-graphs has a vertex of degree at most k, then all its large dilates can be isometrically embedded into any set A⊆R^d of positive upper density, when d>k. We also discuss isometric embeddings of distance graphs into subsets of the integer lattice Z^d.