Even Number Game

The game

Given $k$, consider the following game of $2$ players. In each round,

• player 1 chooses a set $R$ of $k$ natural numbers that he has not yet chosen earlier
• player 2 chooses an even number $n\in[\min R,\max R]$, that he has not yet chosen earlier

If player 2 is unable to make a move at some round, he loses the game; otherwise he wins.

Known facts

Fact. Player 1 wins when $k=2$ or $k=3$.

The problem

Question. Is there $k$ such that player 2 wins?

Relevant contents

• article - Compression of enumerations and gain
• talk - Compression of enumerations and gain
• open problem - $n$-r.e. plus left-r.e.