Given $n$. Two players in turn mark points on the plane. No three may be collinear, no $n$ may form a convex $n$-gon. The player who does not have legal move loses. Who has a winning strategy?



Erdős-Szekeres game
The problem was raised and discussed in Parikshit Kolipaka and Sathish Govindarajan, Two player game variant of the Erdős-Szekeres problem, Discrete Math. Theor. Comput. Sci. 15 (2013), no. 3, 73–100, MR3141828. It says the second player wins for $n=5$. 


