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Thread: Existence of constant [TEX]f[/TEX]

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    MHF Contributor Also sprach Zarathustra's Avatar
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    Existence of constant [TEX]f[/TEX]

    Hello!

    Please help me prove the following,

    Let $\displaystyle n\in\mathbb{N}$. Show that there exist infinite $\displaystyle B\subseteq\mathbb{N}-\{n \}$ such that $\displaystyle f$ is constant on all sets of the form $\displaystyle \{n,b\}$ where $\displaystyle b\in B$.


    Thank you!
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    Re: Existence of constant [TEX]f[/TEX]

    What is f here?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Existence of constant [TEX]f[/TEX]

    Oh, sorry!

    Edited!

    $\displaystyle f:[\mathbb{N}]^2\to \{0,1 \}$
    Last edited by Also sprach Zarathustra; Nov 14th 2012 at 02:49 PM.
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    Re: Existence of constant [TEX]f[/TEX]

    Are you proving this for some particular f or for all f?
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Existence of constant [TEX]f[/TEX]

    Makarov, I'm trying to prove Ramsey theorem:

    Let $\displaystyle [\mathbb{N}]^2$, set collection in size 2 of naturals. So for all function $\displaystyle f:[\mathbb{N}]^2\to \{ 0,1 \}$ exist infinite $\displaystyle A\subseteq\mathbb{N}$ such that $\displaystyle f$ is constant on all elements $\displaystyle [A]^2$
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    Re: Existence of constant [TEX]f[/TEX]

    Quote Originally Posted by Also sprach Zarathustra View Post
    Let $\displaystyle n\in\mathbb{N}$. Show that there exist infinite $\displaystyle B\subseteq\mathbb{N}-\{n \}$ such that $\displaystyle f$ is constant on all sets of the form $\displaystyle \{n,b\}$ where $\displaystyle b\in B$.
    I have even more questions other than the one in the last reply.
    You say $\displaystyle f:\mathbb{N}^2\to \{0,1 \}$
    I assume by $\displaystyle \mathbb{N}^2$ you mean $\displaystyle \mathbb{N}\times\mathbb{N}$. Is that correct?
    If so, how does "is constant on all sets of the form $\displaystyle \{n,b\}$" work.
    Do you mean an ordered pair, $\displaystyle (n,b)~?$.
    Moreover, do you mean any $\displaystyle n\in\mathbb{N}~?$
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Existence of constant [TEX]f[/TEX]

    Quote Originally Posted by Plato View Post
    I have even more questions other than the one in the last reply.
    You say $\displaystyle f:\mathbb{N}^2\to \{0,1 \}$
    I assume by $\displaystyle \mathbb{N}^2$ you mean $\displaystyle \mathbb{N}\times\mathbb{N}$. Is that correct?
    If so, how does "is constant on all sets of the form $\displaystyle \{n,b\}$" work.
    Do you mean an ordered pair, $\displaystyle (n,b)~?$.

    It was a mistake! I meant to $\displaystyle f:[\mathbb{N}]^2\ to \{0,1\}$, where $\displaystyle [\mathbb{N}]^2$ is a collection of sets in size two, in other words all sets in the form of $\displaystyle \{a,b \}$ where $\displaystyle a,b\in \nathbb{N}$.

    Moreover, do you mean any $\displaystyle n\in\mathbb{N}~?$
    We choose such $\displaystyle n\in\mathbb{N}$
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    Re: Existence of constant [TEX]f[/TEX]

    Quote Originally Posted by Also sprach Zarathustra View Post
    It was a mistake! I meant to $\displaystyle f:[\mathbb{N}]^2\ to \{0,1\}$, where $\displaystyle [\mathbb{N}]^2$ is a collection of sets in size two, in other words all sets in the form of $\displaystyle \{a,b \}$ where $\displaystyle a,b\in \mathbb{N}$.
    We choose such $\displaystyle n\in\mathbb{N}$
    Let's fix $\displaystyle n$.
    Define $\displaystyle \mathcal{A}=\{k\in\mathbb{N}\setminus\{n\}:f(\{n,k \})=1\}.$

    If $\displaystyle \mathcal{A}$ is infinite, then how would we define $\displaystyle \mathcal{B}~?$

    If If $\displaystyle \mathcal{A}$ is finite, then how would we define $\displaystyle \mathcal{B}~?$
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