Erdos–Szekeres Theorem

The Erdős–Szekeres Theorem deals with sequences and patterns. It states that for any sequence of numbers, if the sequence is long enough, it must contain a smaller increasing or decreasing subsequence of a certain length. Specifically, in any sequence of more than \( mn \) numbers, there will always be an increasing subsequence of length at least \( m+1 \) or a decreasing subsequence of length at least \( n+1 \). This theorem helps us understand how order and structure naturally emerge in large sets of data, illustrating that chaos gives way to recognizable patterns when the set is sufficiently large.