Van der Waerden number is a big fancy term that mathematicians use to describe how many kids you need to have in a room so that you can always find a group of them who are wearing the same color shirt or who are standing in order according to their height.
Imagine you have a bunch of kids in a room and each one of them is wearing a shirt that is either red or blue. You want to know how many kids you need to have in the room so that you can always pick out a group of them who are all wearing red shirts or a group who are all wearing blue shirts. This is where van der Waerden number comes in.
In math terms, the van der Waerden number is the minimum number of kids (let's call it "n") you need to have in a room so that no matter how you color their shirts, you can always find a group of "k" kids who are all wearing the same color. "K" is a number that is smaller than "n", and it could be any number you choose.
For example, let's say "k" is equal to 3. This means you want to find a minimum number "n" of kids so that no matter how you color their shirts, you can always find a group of 3 kids who are all wearing the same color. The van der Waerden number for this case is 9, which means if you have 9 kids in the room, you can always find a group of 3 kids who are wearing the same color.
But why is this important? Well, van der Waerden number helps mathematicians study patterns and structures in many different areas, like computer science and number theory. It also helps in solving problems related to logic and puzzles.
So, there you have it! Van der Waerden number is a fancy math term that tells us how many kids we need to have in a room to always find a group who are wearing the same color shirt or who are standing in order according to their height.