18.3.2. Pigeonhole Principle Argument

The claim is that irrespective of the distinct real numbers that you have in your sequence, you always have a subsequence of length k + 1 which is either strictly increasing or strictly decreasing.

The pigeonhole principle states that if you want to place more items than there are containers (pigeonholes), at least one container must contain more than one item. In this context, we are dealing with distinct real numbers. The claim suggests that when we take any sequence of k + 1 distinct real numbers, there will always be a subsequence of length k + 1 that is either strictly increasing or strictly decreasing. This sets the stage for the proof using the pigeonhole principle.

Imagine you have a set of 10 distinct colored balls and you want to arrange them in a line. According to the pigeonhole principle, if you group these balls by color (pigeonholes), you will eventually be forced to have at least one group that has more than one ball, emphasizing similar characteristics within your selections.

A strictly increasing sequence is a sequence of the form (a_1, a_2, …) where a_1 < a_2 < a_3 < … and a strictly decreasing sequence is defined similarly as a sequence where a_1 > a_2 > a_3 > …

To understand the concept, we first need to define what we mean by strictly increasing and strictly decreasing sequences. A strictly increasing sequence is one where each number is larger than the last, for example, (1, 2, 3). Conversely, a strictly decreasing sequence is one where each number is smaller than the last, such as (3, 2, 1). Understanding these definitions is crucial for the later parts of the argument.

Think of increasing sequences like climbing stairs—you keep moving up to higher steps. Likewise, decreasing sequences are like going down a slide—you continually lower your height as you descend.

A subsequence means that the values may not be consecutive. For example, from a sequence 1, 3, 0, -5, 2, 8, a possible subsequence could be (1, 2, 8).

A subsequence is derived from a sequence but doesn't require all elements to be taken in order. For instance, if we have the sequence 1, 3, 0, -5, 2, 8, we can choose the numbers 1, 2, and 8, thus skipping the others. The flexibility to choose non-consecutive numbers is what makes subsequences interesting as it allows various combinations.

Imagine you're picking ingredients for a salad from a huge grocery list. You might skip some items and just choose the ones you want (carrots, lettuce, cucumbers) to create a fabulous salad—this selection are the subsequences you form from your complete list.

To prove the statement, we let the arbitrary sequence of k + 1 distinct real numbers be denoted by a_i.

To prove the pigeonhole principle’s claim, we start with an arbitrary sequence of k + 1 distinct real numbers. The essence of the proof involves identifying pairs of increasing and decreasing subsequences for each number in the sequence, linking them through their lengths.

Think of it as painting a picture where each stroke is a different color. You keep adding colors (the distinct real numbers), and based on how you mix them, you notice a pattern emerging—certain colors will interact and define boundaries (increasing or decreasing sequences) just like subsequences emerge from your larger collection.

Assume that for each a_i in the sequence, the length of the longest increasing subsequence is at most k, leading to a contradiction via the pigeonhole principle.

In this part, we assume the opposite of what we want to prove: that every increasing or decreasing subsequence does not exceed length k. By pairing these lengths, we find more pairs than possible unique values based on the pigeonhole principle, leading to a contradiction where at least one subsequence must breach this limit.

Imagine folding a long paper into sections. If you keep folding the paper but have more sections than folds, inevitably some sections will overlap, showing us that there are more subsequences than the limits imposed by the initial folds—hence, contradiction.

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