On colorings of finite projective planes

✍ Scribed by Charles H. Jepsen

• Publisher: Elsevier Science
• Year: 1987
• Tongue: English
• Weight: 125 KB
• Volume: 63
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(87)90155-5

✦ Synopsis

Using counting arguments we extend previous results concerning the coloring of lines in a finite projective plane of order n whose points are n-colored.

Suppose the points of the finite projective plane PG(2, n) are colored with n colors. Kabell [2] showed that at least one line must contain points of at most n -1 colors. Csima [1] strengthened this for planes of odd order by showing that at least one line must have three points of the same color.

In this note, we improve Kabell's result by proving that, for n >13, any n-coloring of PG(2, n) must produce at least n lines containing points of at most n-1 colors. In addition, we exhibit a coloring where the lower bound n is attained. Observe first that the condition n I> 3 is necessary; indeed, it is easy to find 2-colorings of PG(2, 2) for which there is only one monochromatic line.

Before presenting the theorem, we give two preliminary lemmas.

Lemma 1. Suppose that m points of PG(2, n) are colored red, where m <~ n. Then at least n lines contain no red points.