Colouring finite incidence structures (Q1091607)

Most of the results of this paper are concerned with colouring finite projective planes P of order n. Here are two of the results. (i) If \(\chi <(n^ 2+n+1)/3,\) then any \(\chi\)-colouring of P results in at least one line having \(n-1\) or fewer colours. (ii) If no colour appears more than twice on any line in a \(\chi\)-colouring of P, then \(\chi \geq n+1.\) There is also a result concerned with colouring finite affine planes A of order n. (iii) If no colour appears more than twice on any line in a \(\chi\)- colouring of A, then \(\chi \geq n\quad (n\quad odd),\chi \geq n-1\) (n even).