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I'm a first year Computer Science student and I've just finished the Algebra I course. This included an introduction to set theory, equivalence relations, countable sets, factor set, groups, (iso)morphism. During my study I've stumbled upon the Cantor-Bernstein Theorem:

Consider two sets $A$ and $B$. If there exists two injective functions $f:A\rightarrow B$ and $g:B\rightarrow A$, then $|A|=|B|$.

I've been trying to prove this theorem by using graph theory, by considering that the elements of $A$ and $B$ are the vertexes of a oriented bipartite graph and the two functions $f$ and $g$ determine a series of one-way arches between these vertexes.

After some time I realized that the two sets may as well be uncountable, thus making finite graph theory of no use. I'd like to ask anyone here for some recommendations on infinite graph theory (pdfs, books, lecture notes etc.).

Thanks in advance!

PS: All my knowledge in graph theory comes from my work with them as a programming-related subject, not math-related.

S.C.B.
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theSongbird
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    Since graphs do not have to satisfy a Cantor--Bernstein property (i.e. you can find two graphs, each embeds into the other, but they are not isomorphic). I don't know what you would gain by looking for a graph theoretic proof. It is true that you can probably think about the orbits of elements under $f\circ g$ as they move around. But thinking about that from a graph theoretic perspective might not be helpful at all. Not to mention that once you move from finite to infinite, set theory as a whole becomes a much more important player in the game. So maybe first you should learn set theory. – Asaf Karagila Feb 14 '17 at 14:33
  • @AsafKaragila Thanks for the advice! I was thinking about proving the theorem from a graph perspective as a leisure time activity (or "for fun", you may say). – theSongbird Feb 14 '17 at 14:38
  • Here's a reference for infinite graphs: https://link.springer.com/chapter/10.1007/978-3-662-53622-3_8 (I'm not a graph theorist, so I can't say if it's the best one.) Also, there is a way to prove the Cantor-Bernstein theorem using graph theory; see this question and its answer: https://math.stackexchange.com/questions/2608367/cantor-bernstein-schr%C3%B6der-theorem-small-proof-using-graph-theory-is-this-corre – Trevor Wilson Jan 15 '21 at 04:27

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