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How do IMO Problems Compare with Research Problems?

Ramsey Theory as a Case Study

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An Invitation to Mathematics

Abstract

Although IMO contestants and research mathematicians are both attempting to solve difficult mathematical problems, there are important differences between their two activities. This is partly because most research problems involve university-level mathematical concepts that are excluded from IMO problems. However, there are more fundamental differences that are not to do with subject matter. To demonstrate this, we look at some results and questions in Ramsey theory, an area that has been a source both of IMO problems and of research problems.

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Further Reading

  1. Ron Graham, Bruce Rothschild, and Joel Spencer, Ramsey Theory. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1990). This book contains a wealth of material about Ramsey’s theorem, van der Waerden’s theorem, Hindman’s theorem, and many other results. It is a highly recommended starting point for anyone interested in the subject.

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  2. Béla Bollobás, Linear Analysis: An Introductory Course, second edition. Cambridge University Press, Cambridge (1999), xii+240 pp. Banach spaces belong to a branch of mathematics known as Linear Analysis. This is an introduction to that area that is likely to appeal to IMO contestants. (Look out for the exercise that has two stars …)

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  3. Edward Odell and Thomas Schlumprecht, The distortion problem. Acta Mathematica 173, 259–281 (1994). This publication contains the example of Odell and Schlumprecht that is mentioned in Section 5.

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  4. W. Timothy Gowers, An infinite Ramsey theorem and some Banach-space dichotomies. Annals of Mathematics (2) 156, 797–833 (2002). This publication contains my result about the infinite game and its consequences.

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  5. W. Timothy Gowers, Ramsey methods in Banach spaces. In: William B. Johnson and Joram Lindenstrauss (editors), Handbook of the Geometry of Banach Spaces, volume 2, pp. 1071–1097. North-Holland, Amsterdam (2003)

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  6. William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces. In: William B. Johnson and Joram Lindenstrauss (editors), Handbook of the Geometry of Banach Spaces, volume 1, pp. 1–84. North-Holland, Amsterdam (2001)

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Correspondence to W. Timothy Gowers .

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© 2011 Springer-Verlag Berlin Heidelberg

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Gowers, W.T. (2011). How do IMO Problems Compare with Research Problems?. In: Schleicher, D., Lackmann, M. (eds) An Invitation to Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19533-4_5

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