MDOLLS on SPOJ uses Dilworth's theorem.

The classical O(N lg N) algorithm for longest increasing subsequence (LIS) can be seen as an application of Dilworth's Theorem. See here: http://www.geeksforgeeks.org/longest-monotonically-increasing-subsequence-size-n-log-n/

A problem from the third round of 2015 Facebook Hacker Cup

https://www.facebook.com/hackercup/problem/847639175277938/

Solution: https://www.facebook.com/notes/1056536891028878

Another good problem (Dilworth's on longest increasing subsequence) is Cow Jog from USACO December 2014.