Density version of the Ramsey problem and the directed Ramsey problem

We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on $n$ vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges $|E_{RB}|$ is given. The aim is to find the maximal size $f$ of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on $n$ vertices such that the number of bi-oriented edges $|E_{bi}|$ is given. The aim is to bound the size $F$ of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if $|E_{RB}|=|E_{bi}| = p{n\choose 2}$, ($p\in [0,1)$), then $f, F < C_p\log(n)$ where $C_p$ only depend on $p$, while if $m={n \choose 2} - |E_{RB}| <n^{3/2}$ then $f= \Theta (\frac{n^2}{m+n})$. The latter case is strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| = p{n\choose 2}

The Minimum Feedback Arc Set Problem is NP-hard for Tournaments

International audienceAnswering a question of Bang-Jensen and Thomassen, we prove that the minimum feedback arc set problem is NP-hard for tournaments

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 $H$