Width of a product of chains 
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*Write $E_k = (\{0, \ldots, k-1\}, \leq)$ the total order on $k$ elements.

*If $(A, \leq_A)$ and $(B, \leq_B)$ are two posets, the product poset is $(A \times B, \leq_{A \times B})$ with the order defined by $(a, b) \leq_{A \times B} (a', b')$ iff $a \leq_A a'$ and $b \leq_B b'$. This product is associative.

*An antichain of a poset $(X, \leq)$ is a subset $A \subseteq X$ such that all elements of $A$ are pairwise incomparable. The width of a poset is the cardinality of its largest antichain.


Given $x_1, \ldots, x_n$, what is the width of $E_{x_1} \times \cdots \times E_{x_n}$?
Simple examples: the width of $E_k$ is $1$ for any $k$, the width of $E_k^2$ is $k$ (becasue $\{(0, k), (1, k-1), \ldots, (k, 0)\}$ is an antichain and the pidgeonhole principle ensures that a larger subset would necessarily contain two comparable elements).

This is another related paper:
Jerrold R. Griggs Maximum antichains in the product of chains.
