K\"onig's theorem says that in every bipartite graph $G=(U,V; E)$, the size of a maximum matching (a collection of disjoint edges) equals the size of a minimum vertex cover (a collection of vertices which intersects every edge).  If the size of the matching is finite, this is a non-trivial result.  However, if the size of the matching is infinite, this result is trivial.  

Erd\"os suggested the following reformulation of K\"onig's theorem: in every bipartite graph $G=(U,V; E)$ there exists a matching $M$ and a vertex cover $C$ such that for all $e\in M$, $|e\cap C|=1$.  He conjectured that this reformulated version of K\"onig's theorem is true for all infinite bipartite graphs.  

Aharoni gave a highly non-trivial proof of Erd\"os' conjecture (<cite authors="Aharoni, Ron">[**Koenig’s duality theorem for infinite bipartite graphs**](http://dx.doi.org/10.1112/jlms/s2-29.1.1), J. Lond. Math. Soc., II. Ser. 29, 1-12 (1984). [ZBL0505.05049](https://zbmath.org/?q=an:0505.05049).</cite>).

An analogous situation exists for Menger's theorem, so I'm not sure if it's worth spelling it all out, but I will say that the proof of the infinite version of the reformulated Menger's theorem was finally completed in a 62 page paper of Aharoni and Berger (<cite authors="Aharoni, Ron; Berger, Eli">[**Menger’s theorem for infinite graphs**](http://dx.doi.org/10.1007/s00222-008-0157-3), Invent. Math. 176, No. 1, 1-62 (2009). [ZBL1216.05092](https://zbmath.org/?q=an:1216.05092).</cite>).