  An $r$-uniform hypergraph ($r$-graph for short) is called linear if every
pair of vertices belong to at most one edge. A linear $r$-graph is complete if
every pair of vertices are in exactly one edge. The famous
Brown-Erd\H{o}s-S\'os conjecture states that for every fixed $k$ and $r$, every
linear $r$-graph with $\Omega(n^2)$ edges contains $k$ edges spanned by at most
$(r-2)k+3$ vertices. As an intermediate step towards this conjecture, Conlon
and Nenadov recently suggested to prove its natural Ramsey relaxation. Namely,
that for every fixed $k$, $r$ and $c$, in every $c$-colouring of a complete
linear $r$-graph, one can find $k$ monochromatic edges spanned by at most
$(r-2)k+3$ vertices. We prove that this Ramsey version of the conjecture holds
under the additional assumption that $r \geq r_0(c)$, and we show that for
$c=2$ it holds for all $r\geq 4$.
