Ramsay problems for spaces with symmetries

For a wide class of spaces endowed with symmetries, the maximum size of a one-colour symmetric set existing under any colouring of the space in a given number of colours is determined. Discrete spaces (finite Abelian groups and initial segments of the positive integers) and continuous algebraic and geometric structures (a closed interval on the real line, figures of revolution, and compact Abelian groups) are investigated.