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

    f:[\mathbb{N}]^2\to \{0,1 \}
    Last edited by Also sprach Zarathustra; November 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 [\mathbb{N}]^2, set collection in size 2 of naturals. So for all function f:[\mathbb{N}]^2\to \{ 0,1 \} exist infinite A\subseteq\mathbb{N} such that f is constant on all elements [A]^2
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    Re: Existence of constant [TEX]f[/TEX]

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

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

    Moreover, do you mean any n\in\mathbb{N}~?
    We choose such 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 f:[\mathbb{N}]^2\ to \{0,1\}, where [\mathbb{N}]^2 is a collection of sets in size two, in other words all sets in the form of \{a,b \} where a,b\in \mathbb{N}.
    We choose such n\in\mathbb{N}
    Let's fix n.
    Define \mathcal{A}=\{k\in\mathbb{N}\setminus\{n\}:f(\{n,k  \})=1\}.

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

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