We show that the size-Ramsey number of any cubic graph with n $n$ vertices is O ( 8 / 5 ) $O(n^{8/5})$ , improving a bound 3 + o 1 $n^{5/3 o(1)}$ due to Kohayakawa, Rödl, Schacht, and Szemerédi. The heart argument there constant C $C$ such random $C n$ where every edge chosen independently probability p ⩾ − 2 $p \geqslant n^{-2/5}$ high Ramsey for vertices. This latter result best possible up c...

The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest edges $G$ can have, such that for every edge-coloring with $r$ colors there exists monochromatic copy in $G$. notion size-Ramsey numbers has been introduced by Erdős, Faudree, Rousseau and Schelp 1978, attracted lot attention ever since. For $H$, we denote $H^q$ obtained from subdividing its $q{-}1$ vertices each. In recen...

For a k-uniform hypergraph G with vertex set {1, . . . , n}, the ordered Ramsey number ORt(G) is the least integer N such that every t-coloring of the edges of the complete k-uniform graph on vertex set {1, . . . , N} contains a monochromatic copy of G whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual gr...