Essentially infinite colourings of hypergraphs

Abstract

We consider edge colourings of the complete r-uniform hypergraph K (r) n on n vertices. How many colours may such a colouring have 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 (r) n receives at most k different colours. We investigate the threshold for k that guarantees that every (H, k)-local colouring of K (r) n must have a globally bounded number of colours as n → ∞, and we establish this threshold exactly. The following phenomenon is also observed: for many H (at least in the case of graphs), if k is a little over this threshold, the unbounded (H, k)-local colourings exhibit their colourfulness in a “sparse way”; more precisely, a bounded number of colours are dominant while all other colours are rare. Hence we study the threshold k0 for k that guarantees that every (H, k)-local colouring γn of K (r) n with k ≤ k0 must have a globally bounded number of colours after the deletion of up to enr edges for any fixed e > 0 (the bound on the number of colours is allowed to depend on H and e only); we think of such colourings γn as “essentially finite”. As it turns out, every essentially infinite colouring is closely related to a non-monochromatic canonical Ramsey colouring of Erdős and Rado. This second threshold is determined up to an additive error of 1 for every hypergraph H. Our results extend earlier work for graphs by Clapsadle and Schelp [Local edge colorings that are global, J. Graph Theory 18 (1994), no. 4, 389–399] and by the first two authors and Schelp [Essentially infinite colourings of graphs, J. London Math. Soc. (2) 61 (2000), no. 3, 658–670]. We also consider a related question for colourings of the integers and arithmetic progressions.