The upper chromatic number of PG(2,q) and small double blocking sets
Suppose you want to color the points of PG(2,q) with as many colors as
possible without creating a rainbow line (that is a line whose points have
pairwise distinct colors). A easy way to do it is to paint a smallest double
blocking set entirely red and color all other points with mutually distinct
colors. We will show that this is the best possible coloring (under some
conditions on q, e.g., q not a prime). In order to show this, we use
combinatorial and algebraic tools, and in PG(2,p^h), h>2 odd, p odd, we
construct two disjoint blocking sets of size p^h+p^(h-1)+...+p+1. These are
joint results with Gabor Bacso and Tamas Szonyi.