Timeline for Games and Ramsey's theorem
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2019 at 0:51 | comment | added | user78375 | Since whether or not a sequence $(m_n, l_n)_{n=1}^\infty$ lies in $\mathcal{V}$ depends only on its initial segments, I would like to know if it is possible to choose the $l_n$ sequence in a game without future information. That is, if every $A$ has a subset homogeneous in $\mathcal{V}$, is it possible to write down game where Players take turns choosing infinite subsets and $m_n, l_n$ values, and one of the players wins if $(m_n, l_n)_{n=1}^\infty\in \mathcal{U}$ (equivalently, $(m_n, l_n)_{n=1}^t\in \mathcal{F}$ for all $t$? Basically, can we choose the $l_n$ concurrently with the $m_n$? | |
| Jun 24, 2019 at 0:47 | comment | added | user78375 | This is about as far as I had gotten. The hypothesis that for every infinite subset $K$ of $\mathbb{N}$, there exists a further subset $M$ which is homogeneous in $\mathcal{V}$. If we write $M=(m_n)_{n=1}^\infty$, this means there exists $(l_n)_{n=1}^\infty$ such that $(m_n, l_n)_{n=1}^t\in \mathcal{F}$ for all $t$. However, this is only good enough to let me choose $(l_n)_{n=1}^\infty$ after all $(m_n)_{n=1}^\infty$ have been chosen. | |
| Jun 24, 2019 at 0:27 | history | answered | Gabriel Medina | CC BY-SA 4.0 |