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

Abstract

The size-Ramsey number $\^{r} $(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 $\^{r} $(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, $\^{r} $(Pn) < 137n for n sufficiently large.