As Darij said in the comments, Dickson's lemma and its variants often come up in Gröbner theory.  One application of this is Sam and Snowden's [proof][1] of the Noetherian property for various categories (like $FI$) which arise in representation stability.

As an immediate corollary one recovers the eventual stability of Kronecker coefficients (structure constants for tensor products of symmetric group representations).


  [1]: https://arxiv.org/abs/1409.1670v3