The semi-random method was introduced in the early eighties. In its first form of the method lower bounds were given for the size of the largest independent set in hypergraph with certain uncrowdedness properties. The first geometrical application was a major achievement in the history of Heilbronn's triangle problem. It proved that the original conjecture of Heilbronn was false. The semi-random method was extended and applied to other problems. In this talk we give two further geometrical applications of it. First we give a slight improvement on Payne and Wood's upper bounds on a Ramsey-type parameter, introduced by Gowers. We prove that if given any planar point set of size Ω(n2lognloglogn) , then one can find n points on a line or n independent points. Second we give a slight improvement on Schmidt's bound on Heilbronn's quadrangle problem. We prove that there exists a point set of size n in the unit square that doesn't contain four points with convex hull of area at most O(n−3/2(logn)1/2) . This is a joint work with Péter Hajnal.

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