  Erd\H{o}s and Pach (1983) introduced the natural degree-based generalisations
of Ramsey numbers, where instead of seeking large monochromatic cliques in a
$2$-edge coloured complete graph, we seek monochromatic subgraphs of high
minimum or average degree. Here we expand the study of these so-called
quasi-Ramsey numbers in a few ways, in particular, to multiple colours and to
uniform hypergraphs.
  Quasi-Ramsey numbers are known to exhibit a certain unique phase transition
and we show that this is also the case across the settings we consider. Our
results depend on a density-biased notion of hypergraph discrepancy optimised
over sets of bounded size, which may be of independent interest.
