In this talk we will sketch some combinatorial statements which are quite trivial in the finite case and discuss their infinite analogs, which are quite often way harder (or mabye even false...). After giving some interesting examples in various subareas, we will turn our focus into one specific open problem which is known as the ``unfriendly partition conjecture''. More specifically, given a graph $G=(V,E)$, a partition $V=V_0\cup V_1$ is said to be ``unfriendly'' if every vertex $v\in V_i$ has at least as many neighbors in $V_{i+1}$ rather in $V_i$ (the computation $i+1$ is done mod 2). The statement ``every finite graph has an unfriendly partition'' is trivial. We will see that for the infinite case, this statement can be wrong as was shown by Shelah and Milner, and basically the only unknown case is when $G$ is a graph of an infinite but countable size.

The talk is based on few papers by other researchers (among other: a paper by Soukup, the Shelah-Milner paper, a paper by Thomassen, and more).