The maximum chromatic number of the disjointness graph of segments on n-point sets in the plane with n≤16

Let P be a finite set of points in general position in the plane. The disjointness graph of segments D ( P ) of P is the graph whose vertices are all the closed straight line segments with endpoints in P , two of which are adjacent in D ( P ) if and only if they are disjoint. As usual, we use χ ( D ( P ) ) to denote the chromatic number of D ( P ), and use d ( n ) to denote the maximum χ ( D ( P ) ) taken over all sets P of n points in general position in the plane. In this paper, we show that d ( n ) = n - 2 if and only if n ∈ { 3 , 4 , … , 16 } .