If I'm not mistaken, the original application of the Pigeonhole Principle - the reason we call it Dirichlet's Pigeonhole Principle - was to Dirichlet's Theorem on Diophantine Approximation, viz., if $\alpha$ is a real irrational then there are infinitely many rationals $p/q$ such that $|\alpha-(p/q)|<1/q^2$. An oldie, but still a goodie.