An Alternative Proof of the Linearity of the Size-ramsey Number of Paths

The size-Ramsey number r̂(F ) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that every colouring of the edges of G with two colours yields a monochromatic copy of F . In 1983, Beck provided a beautiful argument that shows that r̂(Pn) is linear, solving a problem of Erdős. In this note, we provide another proof of this fact that actually gives a better bound, namely, r̂(Pn) < 137n for n sufficiently large.