Finite coloring of $[\omega]^{<\omega}$.

Question

It is known that Ramsey theorem does not hold for finite colorings of $[\omega]^{<\omega}$.

So I am interested in this "partial" result:

First let $S_n = ]n, +\infty[$ be the set of natural numbers from $n+1$ to infinity.

Now let

$$c: [\omega]^{<\omega} \rightarrow \{0, 1, ..., k\}$$

be a finite coloring.

Then there are $J, A \in [\omega]^\omega$ and a color $i < k+1$ such that

$$c(\bigcup_{n \in J} [A \cap S_n]^n) = i.$$

I think I should handle to use the finite Ramsey theorem, but I can't see how.

Any hint or solution?

Thank you.

Answer 1

Using the usual Ramsey theorem, inductively construct $\{(n_k, A_k, i_k) : k \in \omega \}$ such that $A_k \in [\omega]^{\omega}$ are decreasing, $n_k = \min(A_k)$ and $c[[A_{k+1}]^{n_{k}}] = \{i_k\}$. Choose $J \in [\omega]^{\omega}$ such that $\{i_k : k \in J\} = \{i_{\star}\}$. Put $A = \{n_k : k \in J\}$ and check. Notice that this heavily uses that your $S_n$ does not contain $n$.