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Abstract. In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N (n) such that any set of at least N (n) points in general position in the plane contains n points that are the vertices of a convex n-gon. They also posed the problem to determine the value of N (n) and conjectured that N (n) = 2 n−2 + 1 for all n ≥ 3.Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting.

This is a review of various problems and results on the illumination of convex bodies in the spirit of combinatorial geometry. The topics under review are: history of the Gohberg-Markus-Hadwiger problem on the minimum number of exterior sources illuminating a convex body, including the discussion of its equivalent forms like the minimum number of homothetic copies covering the body; generalization of this problem for the case of unbounded convex bodies; visibility and inner illumination of convex bodies; primitive illuminating systems for convex bodies; illumination and visibility of families of convex bodies; clouds formed by translates or homothetic copies of a convex body; miscellaneous results.Mathematics Subject Classification (1991). 52A37, 52A40.

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