Undergraduate Math Club Seminar

GLC B122

Among any 5 points in general position in the plane, some 4 points form a convex quadrilateral. This raises the question: how large must n be so that, given any n points in general position in the plane, some k points form a convex polygon? In a classical result, Erdős and Szekeres proved that at least 2^(k-2) + 1 points are necessary. We survey the known results, from the original upper and lower bounds given by Erdős and Szekeres, to a recent result by Suk (2017) which vastly improves on the previous upper bounds.