Colourings with fixed congruence in finite projective planes (Q1365038)

The author investigates nontrivial partitions of the point set of a finite projective plane of order \(q\) such that each partition class meets each line in \(d \bmod h\) points, for some fixed natural numbers \(d\) and \(h\), \(h\geq 2\). The main results are that \(h\neq q\), that \(q\) cannot be prime, nor a product of two distinct odd primes, every partition class is a blocking set, and (hence) the number of partition classes does not exceed \(q-1\). The author also investigates the situation for planes of small order. For \(q=4\), a complete classification is easily obtained, for \(q=8, 9, 12, 16, 18\), restrictions on the pairs \((d,h)\) are obtained, along with partial classification results.