On colorings of finite projective planes (Q1086240)

Suppose a finite projective plane PG(2,n) of order \(n\geq 2\) whose points are n-colored e.g. whose points are colored with n different colors. \textit{J. Kabell} [Discrete Math. 44, 319-320 (1983; Zbl 0516.05042)] showed that at least one line must contain points of at most n-1 colors, and \textit{J. Csima} [Discrete Math. 49, 309 (1984; Zbl 0552.05006)] strengthened this result for planes of odd order showing that at least one line must have three points of the same color. In the paper under review, the author improves the Kabell's result proving that, for \(n\geq 3\), at least n lines must contain points of at most n-1 colors. And moreover, it is shown that this bound is best possible since there exist colorings of PG(2,n) where exactly n lines are colored with at most n-1 colors.