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I have been told that Dilworth's Theorem implies Hall's Theorem. I don't seem to be able to show this implication, especially because I haven't done any graph theory in a while! Any help? If possible, I would like a graph theory proof (and not using systems of distinct representative).

I believe the main idea is to order the bipartite graph $G = X \cup Y$ by letting $x<y$ if and only if $x \in X$, $y\in Y$ and there is an edge between $x$ and $y$. Now if I could show that I can assume the longest anti-chain in $G$ contains the whole of $X$, I am done. But I am not sure this is true! I tried to apply Hall's Condition to a possibly "larger" anti-chain which does not contain some elements of X and show this is not possible, but I was unsuccessful.

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  • $\begingroup$ Here's a way to prove Hall's Theorem using Dilworth's Theorem. Assume Dilworth's Theorem is true. Now forget about that and just prove Hall's Theorem. Original joke. $\endgroup$ – Dan Shved Oct 31 '13 at 16:57
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More generally, you can show that, dilworth theorem implies that in bipartite graphs, maximum matching is equal to minimum vertex cover, and vice versa.

As a hint, one way is easy: the complement of a minimum vertex cover is an anti chain. I leave the other side to you!

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  • $\begingroup$ I was thinking about maximum matching and minimum cover because Dilworth's Theorem seems "similar"! Thank you for the hint! $\endgroup$ – user39280 Nov 1 '13 at 7:54

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