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The determination of the number of monotone Boolean functions of n variables (equivalent to the number of antichains on the n-set {1,2,...,n}) is called Dedekind's problem, ...

If n>19, there exists a Poulet number between n and n^2. The theorem was proved in 1965.

A.k.a. the pigeonhole principle. Given n boxes and m>n objects, at least one box must contain more than one object. This statement has important applications in number theory ...

The set theory symbol (aleph) for the cardinal number of a well-orderable infinite set.

The ordinal number of a value in a list arranged in a specified order (usually decreasing).

A three-colorable graph G is a graph with chromatic number chi(G)<=3.

Let y_n be a complex number for 1<=n<=N and let y_n=0 if n<1 or n>N. Then (Montgomery 2001).

A group in which any decreasing chain of distinct subgroups terminates after a finite number.

If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envelopes. Similarly, there must be a pigeonhole with at most a envelopes.

The number of inward directed graph edges from a given graph vertex in a directed graph.