Colouring 4-cycle systems with equitably coloured blocks

A colouring of a 4-cycle system (V,B) is a surjective mapping φ : V → Γ. The elements of Γ are colours and, for each i ∈ Γ, the set Ci = φ−1(i) is a colour class. If |Γ| = m, we have an m-colouring of (V,B). For every B ∈ B, let φ(B) = {φ(x) | x ∈ B}. We say that a block B is equitably coloured if either |φ(B)∩Ci| = 0 or |φ(B)∩Ci| = 2 for every i ∈ Γ. Let F(n) be the set of integersm such that there exists an m-coloured 4-cycle system of order n with every block equitably coloured. We prove that: • min F(n) = 3 for every n ≡ 1 (mod 8), n ≥ 17, F(9) = ∅; • {m | 3 ≤ m ≤ n+31 16 } ⊆ F(n), n ≡ 1 (mod 16), n ≥ 17; • {m | 3 ≤ m ≤ n+23 16 } ⊆ F(n), n ≡ 9 (mod 16), n ≥ 25; • for every sufficiently large n ≡ 1 (mod 8), there is an integer m such that maxF(n) ≤ m. Moreover we show that max F(n) = m for infinite values of n. AMS classification: 05B05.