Another approach on the four color theroem I've tried to understand why every map can be colored with 4 colors and I think I found a reason for that, but I don't know if my statements and conclusions are correct. Could you check it?
First part shows why 5 countries cannot be adjacent to each other
In the second part I tried to prove the theorem using the statement from the first part.
It is not written in a scientific language.
 A: Disclaimer : I have not read fully your presentation, only the first page so far (and quickly), so this is a first comment on your work. Don't hesitate to correct me if I misread.
In this page you state :

In order to disprove this theory, 5 countries need to be adjacent to each other at the same time. This way 4 colors will not be sufficient

This is true, but not sufficient. The graph colouring number is a global property of the graph, here you are only looking at the local case. Using graph theory methodology you are using the fact that the chromatic number of a graph is at least its clique number (the largest number of mutually adjacent vertices) :
$$ \chi(G)\geq \omega(G)$$
However, the colouring number being a global property, you will have some graphs, where there are no 5 vertices all adjacent (which is equivalent in your case of not having 5 countries adjacent), but still requiring 5 colours. For example

This graph is not 4-colorable, you need 5 colours, but its clique number is only 4. It is non-planar obviously, but the fact that it is not 4-colorable does not come from including a clique of size 5. You need other properties of planar graph to prove that they are 4-colorable.
Edit on integer series :
In your second page you talk about integer series, representing the number of adjacent countries for each country. This is known as degree sequence in graph theory. A degree sequence that represents (at least) one graph is called feasible. Be careful this does not means that it represents a maps, for this it would need to represent a planar graph.
As you already found, there are some easy restriction on feasible degree sequence. The feasibility of a degree sequence is solved by the Erdos-Gallai theorem :

A sequence of non-negative integers $d_{1}\geq \cdots \geq d_{n}$ can be represented as the degree sequence of a finite simple graph on $n$ vertices if and only if $d_{1}+\cdots +d_{n}$ is even and
$$\sum^{k}_{i=1}d_i\leq k(k-1)+ \sum^n_{i=k+1} \min (d_i,k)$$
holds for every k in $1\leq k\leq n$.

You should also have a look at the Havel–Hakimi algorithm. I do not know if there are similar partial results for feasible planar degree sequences.
