This question is considerably related to this one.
As a matter of fact, I need to show that:
Using the Compactness Theorem, show that the Infinite Ramsey Theorem implies the following Finite Ramsey Theorem: $\forall k \in > \mathbb{N}$, $\exists n \in \mathbb{N}$ (depending on $k$) such that every finite graph with at least $n$ vertices contains a clique of size $k$ or an independent set of size $k$.
According to @Noah's guidance, I just tried to prove the case with bridging between coloring version of the theory and vicinity-based one (based on cliques and independent sets).
My instructor has confirmed the validity of my proof but just commented on that there is a better approach by contrapositive (that I can't understand how it does work). As he said, I need to assume that the Finite Ramsey Theorem is false, and show that the Infinite Ramsey Theorem is false using the Compactness Theorem (i.e. construct an infinite graph G with no infinite clique and no infinite independent set).
It's confusing to me as I have no idea how one can use the Compactness Theorem to find a contrapositive.
Could you please enlighten me about this method of proof?
Here's the idea. Fix some $n$ which witnesses that Finite Ramsey's Theorem fails - that is, that there are arbitrarily large finite graphs with no size-$n$ cliques or independent sets. In the language of graphs, the sentence $\kappa_n$="$\forall x_1...x_n(\bigvee_{1\le i<j\le n}\neg R(x_i, x_j))$" asserts the nonexistence of a clique of size $n$. Now:
Can you find a similar sentence $\alpha_n$ asserting the nonexistence of an independent set of size $n$?
Can you find a family $\Phi$ of sentences which collectively assert that there are infinitely many vertices?
The idea is that the set $S_n=\{\kappa_n,\alpha_n\}\cup\Phi$ describes graphs with infinitely many vertices and no cliques/independent sets of size $n$. Now, under the assumption that Finite Ramsey's Theorem fails at $n$ as described above, do you see how to argue that $S_n$ has a model, and why the existence of such a model contradicts the Infinite Ramsey's Theorem?