the Erdős–Szekeres theorem

The Erdős–Szekeres theorem states that for any positive integer \( n \), there is a minimum number \( N \) such that any sequence of \( N \) distinct numbers contains either an increasing subsequence of length \( n \) or a decreasing subsequence of length \( n \). In simple terms, in any large enough set of numbers, you'll always find a long enough rising or falling pattern. For example, with 5 numbers, you'll find a 3-number rising or falling sequence. The theorem quantifies this minimum size, demonstrating that orderings become unavoidable and predictable in large datasets.