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!

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

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

Oh, sorry!

**Edited!**

$\displaystyle f:[\mathbb{N}]^2\to \{0,1 \}$

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

Are you proving this for some particular f or for all f?

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 $\displaystyle [A]^2$

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

Quote:

Originally Posted by

**Also sprach Zarathustra** 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 $\displaystyle n\in\mathbb{N}~?$

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

Quote:

Originally Posted by

**Plato** 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 $\displaystyle a,b\in \nathbb{N}$.

Quote:

Moreover, do you mean any $\displaystyle n\in\mathbb{N}~?$

We choose such $\displaystyle n\in\mathbb{N}$

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

Quote:

Originally Posted by

**Also sprach Zarathustra** 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 $\displaystyle \mathcal{B}~?$