On chromatic indices of finite affine spaces

A line-coloring of the finite affine space A G ( n ,  q ) is proper if any two lines from the same color class have no point in common, and it is complete if for any two different colors i and j there exist two intersecting lines, one is colored by i and the other is colored by j . The pseudoachromatic index of A G ( n ,  q ), denoted by ψ ′( A G ( n ,  q )),  is the maximum number of colors in any complete line-coloring of A G ( n ,  q ). When the coloring is also proper, the maximum number of colors is called the achromatic index of A G ( n ,  q ). We prove that ψ ′( A G ( n ,  q )) ∼  q 1.5 n  − 1 for even n , and that q 1.5( n  − 1)  <  ψ ′( A G ( n ,  q )) <  q 1.5 n  − 1 for odd n . Moreover, we prove that the achromatic index of A G ( n ,  q ) is q 1.5 n  − 1 for even n , and we provide the exact values of both indices in the planar case.