Van der Waerden's theorem
Van der Waerden's theorem is like a magic trick for numbers. It says that if you have a lot of numbers and you group them in any way you want, like in groups of two or groups of three or more, there will always be a pattern somewhere.
Let's say you have a big box of different colored toys. You want to group the toys by color, so you put all the red toys in one pile, all the blue toys in another pile, and so on. Van der Waerden's theorem says that no matter how many toys you have or how you group them, there will always be a group somewhere where all the toys have the same color.
This is really cool because it works with any number of groups, not just colors. You could group the toys by shape or size or anything else, and there will always be a group somewhere with an interesting pattern. It's like a game you can play with numbers!
Let's say you have a big box of different colored toys. You want to group the toys by color, so you put all the red toys in one pile, all the blue toys in another pile, and so on. Van der Waerden's theorem says that no matter how many toys you have or how you group them, there will always be a group somewhere where all the toys have the same color.
This is really cool because it works with any number of groups, not just colors. You could group the toys by shape or size or anything else, and there will always be a group somewhere with an interesting pattern. It's like a game you can play with numbers!