The size Ramsey number of a directed path

Given a graph H, the size Ramsey number r_e(H,q) is the minimal
number m for which there is a graph G with m edges such that
every q-coloring of G contains a monochromatic copy of H. We
study the size Ramsey number of the directed path of length n in
oriented graphs, where no antiparallel edges are allowed. We give
nearly tight bounds for every fixed number of colors. For the case of
two colors we show that there are constants c_1,c_2 such that
$$\frac{c_1  n^{2} \log n}{(\log\log n)^3} \leq r_e(P_n,2) \leq c_2
n^{2}(\log n)^2.$$

Joint work with  Michael Krivelevich and Benny Sudakov.