  Given an $r$-edge-coloured complete graph $K_n$, how many monochromatic
connected components does one need in order to cover its vertex set? This
natural question is a well-known essentially equivalent formulation of the
classical Ryser's conjecture which, despite a lot of attention over the last 50
years, still remains open. A number of recent papers consider a sparse random
analogue of this question, asking for the minimum number of monochromatic
components needed to cover the vertex set of an $r$-edge-coloured random graph
$\mathcal{G}(n,p)$.
  Recently, Buci\'{c}, Kor\'{a}ndi and Sudakov established a connection between
this problem and a certain Helly-type local to global question for hypergraphs
raised about 30 years ago by Erd\H{o}s, Hajnal and Tuza. We identify a modified
version of the hypergraph problem which controls the answer to the problem of
covering random graphs with monochromatic components more precisely. To
showcase the power of our approach, we essentially resolve the $3$-colour case
by showing that $(\log n / n)^{1/4}$ is a threshold at which point three
monochromatic components are needed to cover all vertices of a
$3$-edge-coloured random graph, answering a question posed by Kohayakawa,
Mendon\c{c}a, Mota and Sch\"ulke. Our approach also allows us to determine the
answer in the general $r$-edge coloured instance of the problem, up to lower
order terms, around the point when it first becomes bounded, answering a
question of Buci\'c, Kor\'andi and Sudakov.
