Your internship in Professor Ramsey’s physics lab has been amazing. Until, that is, the professor accidentally stepped through a time portal. You’ve got just a minute to jump through the portal to save him before it closes and leaves him stranded in history. Once you’re through it, the portal will close, and your only way back will be to create a new one using the chrono-nodules from your lab. Activated nodules connect to each other via red or blue tachyon entanglement.
Activate more nodules, and they’ll connect to all the other nodules in the area. As soon as a red or blue triangle is created with a nodule at each point, it opens a doorway through time that will take you back to the present. But the colour of each individual connection manifests at random, and there’s no way to choose or change its olor. And there’s one more problem: each individual nodule creates a temporal instability that raises the chances the portal might collapse as you go through it. So the fewer you bring, the better. The portal’s about to close.
What’s the minimum number of nodules you need to bring to be certain you’ll create a red or blue triangle and get back to the present? Pause here if you want to figure it out for yourself. This question is so rich that an entire branch of mathematics known as Ramsey Theory developed from it. Ramsey Theory is home to some famously difficult problems. This one isn’t easy, but it can be handled if you approach it systematically. Imagine you brought just three nodules. Would that be enough? No, for example, you might have two blue and one red connection and be stuck in the past forever.
Would four nodules be enough? No, there are many arrangements here that don’t give a blue or red triangle. What about five? It turns out that there is an arrangement of connections that avoids creating a blue or red triangle. These smaller triangles don’t count because they don’t have a nodule at each corner. However, six nodules will always create a blue triangle or a red triangle. Here’s how we can prove that without sorting through every possible case: Imagine activating the sixth nodule, and consider how it might connect to the other five.
It could do so in one of six ways: with five red connections, five blue connections, or some mix of red and blue. Notice that every possibility has at least three connections of the same colour coming from this nodule. Let’s look at just the nodules on the other end of those same three colour connections. If the connections were blue, then any additional blue connection between those three would give us a blue triangle. So the only way we could get into trouble is if all the connections between them were red. But those three red connections would give us a red triangle.
No matter what happens, we’ll get a red or blue triangle and open our doorway. On the other hand, if the original three connections were all red instead of blue, the same argument still works, with all the colours flipped. In other words, no matter how the connections are coloured, six nodules will always create a red or blue triangle and a doorway leading home. So you grab six nodules and jump through the portal. You were hoping your internship would give you valuable life experience. It turns out that that didn’t take much time.
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