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While I claim to understand the premise of the pigeonhole principle (if $n > k$ then inserting $n$ elements into $k$ boxes results some boxes containing at least two items), I am still quite bad at applying to principle in practice. Therefore I am looking for the resources (books/sites/psets/old course material etc.) to better learn the pigeonhole principle and other combinatorical arguments. The material should preferably contain problems with varying level of difficulty and expose me to the "most common" types of situation in undergrad/grad level in which a combinatorical argument can be used.

Cartesian Bear
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    A nice introductory treatment of such topics is Mathematics of Choice or How to Count Without Counting by Ivan Niven. – Dave L. Renfro Jul 18 '22 at 10:06
  • "Most common" in undergrad/grad level is very subfield-specific in that many subfields just won't need it at all. Do you plan to go to grad school for combinatorics? – Mark S. Jul 18 '22 at 12:09
  • @MarkS. I agree that my specification sounds a bit vague, but I didn't know how to put to words otherwise. I am probably going to specialize in analysis/measure theory, but I would like to have such a mathematical maturity regarding counting arguments/pigeonhole principle that problems such as: https://math.stackexchange.com/questions/2843461/irrational-angle-rotation-argument/2843535#2843535 click near instantly. – Cartesian Bear Jul 18 '22 at 14:32
  • @MarkS. Maybe this post is a bit too demanding and what I really want is multiple examples of how a problem can be rephrased with the boxes and items. It's also possible that the material linked by Dave's is everything I am looking for. – Cartesian Bear Jul 18 '22 at 14:32
  • @SickSeries That example you link doesn't seem like a counting argument at all to me. I'd say it's an analysis application of how convergent sums work. At the very least I think examples of pigeonhole principle wouldn't help you with it. – Mark S. Jul 18 '22 at 14:43
  • @MarkS. IMO you can you pigeonhole principle in the following way: If the radius of the circle is $s$, then you can cover it with by $\frac{s}{\epsilon}$ (rounded up) intervals of length $\epsilon$. Then taking enough iterations of the evolution function $T(x) = x + \alpha \mod 1$ yields that two points are within an $\epsilon$ of each other. I.e. the orbit of an arbitrary starting point is dense if $\alpha$ is irrational. – Cartesian Bear Jul 18 '22 at 14:57

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