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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
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Pigeonhole Principle for infinite case
Suppose $X_n$ are finite sets for any natural integer $n$. let $Y$ be an infinite subset of $\prod_n X_n$. Do there exist $y$ and $y'$ in $Y$ and an infinite subset $S$ of $\mathbb N$ such that $y_n= …