Globally bounded local edge colourings of hypergraphs

We consider edge colourings of K n – the complete r-uniform hypergraph on n vertices. Our main question is: how ‘colourful’ can such a colouring be if we restrict the number of colours locally? The local restriction is formulated as follows: for a fixed hypergraph H and an integer k we call a colouring (H, k)-local, if every copy of H in the complete hypergraph K n picks up at most k different colours. We will investigate the threshold of k which guarantees that every (H, k)-local colouring must have a bounded global number of colours as n tends to infinity.