Recently Joel David Hamkins posted an entry on the Connect Infinity game.
Connect-$\omega$ is Connect Four but played on an $\omega\times n$ grid ($n$ finite)! The above shows $n=6$. The difference is that Connect-$\omega$ proceeds in $\omega$-many moves and the goal is to make a connected sequence in any row, of $\omega$ many coins of your color (you don’t have to fill the whole row, but rather a connected infinite segment of it suffices). A draw occurs when neither or both players achieve their goals.
In the entry, there are proofs which I don't understand. Could someone explain them in detail?
Board size $\omega\times n$, $n$ finite: Neither player has a winning strategy; both players have drawing strategies.
Either player can ensure that there are infinitely many of their coins on the bottom row: they simply place a coin into some far-out empty column, which also blocks a win for the opponent on the bottom row. Next, observe that neither player can afford to follow the strategy of always answering those moves on top, since this would lead to a draw, with a mostly empty board. Thus, it must happen that infinitely often we are able to place a coin onto the second row. This blocks a win for the opponent on the second row. And so on. In this way, either players can achieve infinitely many of their coins on each row, thereby blocking any row as a win for their opponent. So both players have drawing strategies.
I can't see or visualize how IF either player follows the strategy of always answering those moves on top, it would lead to a draw with a mostly empty board. How exactly would this play out?
And why must it happen infinitely often that either player is able to place a coin on the second row, and how does this imply that either players can place infinite many of their coins on each row above the second row?
EDIT: Please see user326210's answer and my summary.