Arithmetic Ramsey Theory
University of Warwick
Is it possible to colour the natural numbers with 3 colours, so that whenever x and y are of the same colour, their sum x + y has a different colour? What if you use more colours? What if instead one requires that whenever x and y are of the same colour, their product x + y has a different colour?
Arithmetic Ramsey theory is a branch of combinatorics which answers these and related questions, by studying patterns which inevitably appear in any finite colouring of the natural numbers. Despite addressing elementary questions, the answers often involve deep ideas and tools from diverse areas of mathematics, such as graph theory, Fourier analysis, geometric group theory, ergodic theory, and number theory (the talk does not assume familiarity with any of these topics!)
Prof . Moreira presents a survey of this fascinating topic, including some simple looking but still open problems.
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