I'm currently working through "Combinatorics of Finite Sets" by Ian Anderson, mostly to improve at a style of mathematics that I've historically been quite bad at, and I find myself wondering why this sort of work actually matters outside of intrinsic interest - the two books I have on this subject are this and Bollobas, and neither of them seem to have so much of a hint at the motivations for studying this other than because we can.

In particular things like Sperner's Theorem (the one about counting antichains, not the one about triangulations of simplices) are neat but I can't see any really interesting consequences of it, either in combinatorics, other areas of mathematics or outside of mathematics altogether. Are there?

I'm aware of the Littlewood-Offord problem, but it doesn't seem any more applicable than Sperner's theorem itself and it's more or less the only application I've found.

I'm not wedded to examples just about Sperner's theorem. I'd be happy with anything "related" - any of the generalisations of it, the Lubell–Yamamoto–Meshalkin inequality, etc.