Biased orientation games

We study biased {\em orientation games}, in which the board is the complete graph $K_n$, and Maker and Breaker take turns in directing previously undirected edges of $K_n$. At the end of the game, the obtained graph is a tournament. Maker wins if the tournament has some property $\mathcal P$ and Breaker wins otherwise. We provide bounds on the bias that is required for a Maker's win and for a Breaker's win in three different games. In the first game Maker wins if the obtained tournament has a cycle. The second game is Hamiltonicity, where Maker wins if the obtained tournament contains a Hamilton cycle. Finally, we consider the $H$-creation game, where Maker wins if the obtained tournament has a copy of some fixed graph $H$

The size Ramsey number of a directed path

Given a graph H, the size Ramsey number re(H,q) is the minimal number m for which there is a graph G with m edges such that every q-coloring of E(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, showing that for every q ≥ 1 there are constants c1 = c1(q),c2 such that c1(q)n 2q (logn) 1/q (loglogn) (q+2)/q ≤ re ( − → Pn,q +1) ≤ c2n 2q (logn) 2. Our results show that the path size Ramsey number in oriented graphs is asymptotically larger than the path size Ramsey number in general directed graphs. Moreover, the size Ramsey number of a directed path is polynomially dependent in the number of colors, as opposed to the undirected case. Our approach also gives tight bounds on re ( − → Pn,q) for general directed graphs with q ≥ 3, extending previous results.