Are there infinitely many $\alpha \times \beta$ Chomp boards where player 2 wins?

Solution 1:

Yes, for every nonzero ordinal $\alpha$ there is an ordinal $\beta$ such that $\alpha \times \beta$ is a second player win (I can't find the source sorry). So far the only pairs that are known are $1 \times 1$ (trivial), $2 \times \omega$, $3 \times \omega^\omega$, and $4 \times \omega^2$.

The situation with five or more rows gets incredibly complicated, and it seems likely that the $\beta$ for 5 rows is a lot bigger than $\omega^\omega$, although bounds could be found by constructively applying the proof of existence.

It seems (from a very rough first guess) that even numbers will have a much smaller $\beta$ than odd numbers.