Extremal and probabilistic problems in order types - Sales, Marcelo Tadeu de Sá Oliveira (2018)
A configuration is a finite set of points in the plane. Two configurations have the same order type if there exists a bijection between them that preserves the orientation of every ordered triple. A configuration A contains a copy of a configuration B if some subset of A has the same order type of B and we denote this by B \\subset A. For a configuration B and a integer N, the extremal number ex(N,B)=max{|A|: B ot \\subset A \\subset [N]^2} is the maximum size of a subset of [N]^2 without a copy of B. We give an upper bound for general and convex cases. A random N-set is a set obtained by randomly choosing N points uniformly and independently in the unit square. A configuration is n-universal if contains all order types in general position of size n. We obtain the threshold for the n-universal property up to a log log factor, that is, we obtain integers N_0 and N_1 with log log N_1=O(log log N_0) such that if N >> N_1 (N << N_0), then a random N-set is n-universal with probability tending to 1 (tending to 0). We also determine a bound for the probability of obtaining a random set without a copy of a fixed configuration.