Existence of constant [TEX]f[/TEX]

**Also sprach Zarathustra** · November 14th 2012, 01:32 PM

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!

**emakarov** · November 14th 2012, 01:45 PM

What is f here?

**Also sprach Zarathustra** · November 14th 2012, 02:10 PM

Oh, sorry!

Edited!

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

**emakarov** · November 14th 2012, 02:14 PM

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

**Also sprach Zarathustra** · November 14th 2012, 02:33 PM

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 $[A]^2$

**Plato** · November 14th 2012, 02:37 PM

Originally Posted by Also sprach Zarathustra

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 $n\in\mathbb{N}~?$

**Also sprach Zarathustra** · November 14th 2012, 02:47 PM

Originally Posted by Plato

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 $n\in\mathbb{N}$

**Plato** · November 14th 2012, 03:52 PM

Originally Posted by Also sprach Zarathustra

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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