On bounded degree graphs with large size-Ramsey numbers

The size-Ramsey number $\hat r(G')$ of a graph $G'$ is defined as the smallest integer $m$ so that there exists a graph $G$ with $m$ edges such that every $2$-coloring of the edges of $G$ contains a monochromatic copy of $G'$. Answering a question of Beck, Rodl and Szemeredi showed that for every $n\geq 1$ there exists a graph $G'$ on $n$ vertices each of degree at most three, with the size-Ramsey number at least $cn\log^{\frac{1}{60}}n$ for a universal constant $c>0$. In this note we show that a modification of Rodl and Szemeredi's construction leads to a bound $\hat r(G')\geq cn\,\exp(c\sqrt{\log n})$.