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C00002 00002	Here is the form of Ramsey's theorem Juan Bulnes should try to prove.
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Here is the form of Ramsey's theorem Juan Bulnes should try to prove.

Consider a countable infinity of vertices, each vertex being connected
to every other by a thread, each thread being red or black.  Prove that
there is an infinite subset of these vertices, such that the connecting
threads within the subset are all of the same color.

Proof: Choose a vertex v1.  Either v1 is connected to ∞ vertices by
red threads or not.  If yes, consider from now on only the vertices
connected to v1 by red threads, and try for a vertex v2 connected to
∞ by red threads.  If you get it try for v3, etc.  Either this construction
can be continued indefinitely, in which case it constructs the desired
set, or not.  If not, we have an infinite subset A of the vertices within
which each vertex is connected to only a finite number by red threads.
In that case, we can build the desired set as follows: suppose we have
a finite set  B(n) all connected by black threads.  Since each vertex of this
set is connected to only a finite number of vertices by red threads and
there are ∞ vertices in A, we can find a vertex to adjoin to B(n) that is
connected to all the members of B(n) by black threads,
thus giving B(n+1).  Continuing this
construction indefinitely proves the theorem.

It is this proof that should be formalized.