UVictoria Discrete Math Seminar: Felix Christian Clemen

A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of n points in the plane. Since then a great variety of extremal problems in finite planar point sets have been studied. In this talk, we examine several such problems, all of which have in common that they can be reduced to the study of the independence number of an auxiliary hypergraph. 

1) What is the size of the largest subset of the n x n grid whose points determine distinct slopes? 

2) What is the size of the largest monotone general position subset of the n x n grid? 

3) Given n points in the plane, how many triangles can be approximate congruent to equilateral triangles?

The results presented are partially joint work with Balogh, Dumitrescu and Liu.