We call a subgraph of an edge-colored graph rainbow, if all its edges have different colors. The anti-Ramsey number G in complete K n , denoted by r ( ) is the maximum colors edge-coloring with no rainbow copy . In this paper, we determine exact value for star forests and double stars approximate linear forests.

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...