  The typical extremal problem asks how large a structure can be without
containing a forbidden substructure. The Erd\H{o}s-Rothschild problem,
introduced in 1974 by Erd\H{o}s and Rothschild in the context of extremal graph
theory, is a coloured extension, asking for the maximum number of colourings a
structure can have that avoid monochromatic copies of the forbidden
substructure.
  The celebrated Erd\H{o}s-Ko-Rado theorem is a fundamental result in extremal
set theory, bounding the size of set families without a pair of disjoint sets,
and has since been extended to several other discrete settings. The
Erd\H{o}s-Rothschild extensions of these theorems have also been studied in
recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families,
and Hoppen, Lefmann and Odermann for vector spaces.
  In this paper we present a unified approach to the Erd\H{o}s-Rothschild
problem for intersecting structures, which allows us to extend the previous
results, often with sharp bounds on the size of the ground set in terms of the
other parameters. In many cases we also characterise which families of vector
spaces asymptotically maximise the number of Erd\H{o}s-Rothschild colourings,
thus addressing a conjecture of Hoppen, Lefmann and Odermann.
