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

1. ## Existence of constant [TEX]f[/TEX]

Hello!

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!

2. ## Re: Existence of constant [TEX]f[/TEX]

What is f here?

3. ## Re: Existence of constant [TEX]f[/TEX]

Oh, sorry!

Edited!

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

4. ## Re: Existence of constant [TEX]f[/TEX]

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

5. ## 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 $\displaystyle [A]^2$

6. ## Re: Existence of constant [TEX]f[/TEX]

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

7. ## Re: Existence of constant [TEX]f[/TEX]

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

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

8. ## Re: Existence of constant [TEX]f[/TEX]

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 $\displaystyle \mathcal{B}~?$