91Ó°ÊÓ

When we talk about subsequences in mathematics, we refer to a sequence derived by removing some or no elements from another sequence, without changing the order of the remaining elements. For example, if we have a sequence \([3, 1, 7, 5]\), some possible subsequences include \([3, 7, 5]\) and \([1, 5]\). The concept of a subsequence is crucial in many problems related to sequences and is especially important in understanding the Erdős–Szekeres theorem.

A subsequence can be formed in different ways, and the simplicity in forming these allows mathematicians to find patterns and rules, such as the one our exercise highlights.

In our given problem about the heights of 101 people, finding a subsequence involves identifying a group of 11 people whose heights form either a strictly increasing or strictly decreasing pattern. We'll rely on this concept to apply the Erdős–Szekeres theorem effectively.

An increasing sequence is a type of subsequence where each term is larger than the previous term. For example, in the sequence \([2, 4, 7, 9]\), each number is greater than the number that comes before it, making it an increasing sequence.

In our problem, we want to find an increasing subsequence of 11 people within a line of 101 different heights. Given the Erdős–Szekeres theorem, with the right parameters, we're guaranteed to find such a subsequence.

To break this down, imagine our sequence is the heights of the people standing in line. By Erdős–Szekeres, which states that for any sequence of length \(n = rs + 1\), there is either an increasing subsequence of length \((r+1)\) or a decreasing subsequence of length \((s+1)\), our task becomes more manageable. For our parameters, \(r = 10\), since \((r+1) = 11\). When we apply \(n = 10 \cdot 10 + 1 = 101\), we see we have fulfilled the conditions of the theorem, thus ensuring our required increasing sequence exists.

A decreasing sequence is the opposite of an increasing sequence; each term is smaller than the previous term. For instance, in the sequence \([9, 7, 4, 2]\), each number is less than the one before it.

Applying this to our height problem, a decreasing sequence of 11 people means we need 11 people whose heights drop progressively from one individual to the next. Per the Erdős–Szekeres theorem, our probability of finding either an increasing or decreasing subsequence is guaranteed.

When we set our parameters \(s = 10\) so that \((s+1) = 11\), we again satisfy the theorem's requirement, because \(n = 10 \cdot 10 + 1 = 101\). Thus, we can be sure that whether we seek an increasing or decreasing sequence of heights among these people, one will definitely exist. This powerful theorem helps simplify what could otherwise be a very complex problem, guiding us directly to the solution.