On a metric generalization of Ramsey's theorem (Q1376051)

An increasing sequence of reals \(x=\{x_i\}\) is simple if all gaps \(x_{i+1}-x_i\) are different. Two simple sequences \(x\) and \(y\) are distance similar if the consecutive distances are ordered in the same way, that is \(x_{i+1}-x_i < x_{j+1}-x_j\) iff \(y_{i+1}-y_i < y_{j+1}-y_j\) for all pairs \(i,j.\) The paper proves that given any bounded simple sequence \(x\) and any colouring of the pairs of rational numbers by finite number of colours, there is always a sequence \(y\) distance similar to \(x\) such that all pairs of \(y\) are of the same colour. A number of analogous results are proved and some interesting counterexamples are given.