Kuratowski–Ulam theorem
Okay kiddo, so imagine you have a bunch of dots on a piece of paper and you can draw lines between those dots. Sometimes, you might want to draw a line that connects every dot to every other dot on the paper. That is called a complete graph. Now, the Kuratowski-Ulam Theorem says that if you try to draw a complete graph on a piece of paper without taking the lines and bending them, then you cannot do it without drawing some extra lines that cross over each other.
Think of it like trying to avoid crossing your arms while hugging your friends all at the same time. It's pretty hard to do and sometimes your arms might end up crossing over each other. That's kind of what happens when you try to draw a complete graph on a piece of paper. You can sometimes end up with extra lines that cross over each other, just like your arms.
The theorem is important because it helps us understand how we can and cannot draw complete graphs without having to bend the lines. It's kind of like a rule to help us figure out if something is possible or not. So if you ever want to draw a complete graph, just remember that the Kuratowski-Ulam Theorem says you might have to bend some lines or add extra lines that cross over each other.
Think of it like trying to avoid crossing your arms while hugging your friends all at the same time. It's pretty hard to do and sometimes your arms might end up crossing over each other. That's kind of what happens when you try to draw a complete graph on a piece of paper. You can sometimes end up with extra lines that cross over each other, just like your arms.
The theorem is important because it helps us understand how we can and cannot draw complete graphs without having to bend the lines. It's kind of like a rule to help us figure out if something is possible or not. So if you ever want to draw a complete graph, just remember that the Kuratowski-Ulam Theorem says you might have to bend some lines or add extra lines that cross over each other.