SYMMETR Y AND COLORINGS: SOME RESUL TS AND OPEN PROBLEMS

In this paper, we survey some principal results and open problems related to colorings of algebraic and geometric objects endowed with symmetries. Starting in 1995 with seminal questions posed by I.V. Protasov, the topic has evolved into a unique field within Ramsey theory characterized by specific methods and a host of complex problems, demonstrating connections to various mathematical disciplines including combinatorial set theory, group theory, topology, geometry, and probability theory. A central question we explore is the identification of the maximal size of a monochromatic symmetric subset within arbitrary 'good' colorings of a given space X endowed with symmetries. This inquiry can often be approached from two perspectives: cardinality and measure, with the results converging on finite objects. Our investigation highlights considerable progress achieved over the last five years while simultaneously acknowledging the existence of numerous intriguing unsolved problems that remain in this area. We hope this survey will stimulate further research and exploration in these dimensions of symmetry and colorings.