  The size-Ramsey number of a graph $F$ is the smallest number of edges in a
graph $G$ with the Ramsey property for $F$, that is, with the property that any
2-colouring of the edges of $G$ contains a monochromatic copy of $F$. We prove
that the size-Ramsey number of the grid graph on $n\times n$ vertices is
bounded from above by $n^{3+o(1)}$.
