I’d give up chocolate, but I’m no quitter.
Welcome back to UniScoops! We’re like a good Wi-Fi signal in the library basement - rare, precious, and life-saving.
Here’s a taste of what we’re serving today:
Ramsey theory 🧮
PLUS: Can Video Games Improve Learning?, Global Tipping Points, and Are We Alone in the Universe? 👽️
Ramsey Theory 🧮
It is a frequently discussed topic among undergraduate mathematicians at lunch: "what is your favourite proof?" There are a few reasons why a proof might be someone's favourite. One reason might be because it can be explained to someone with even the most basic maths education. We call these proofs elementary, not because they're easy but because they don't require much prior knowledge.
I will introduce you to a problem in a field called Ramsey Theory (named after Frank Ramsey, a brilliant mathematician who tragically died at a young age of 26). Ramsey Theory, to oversimplify a century-old field, seeks to find out the size of a group of things that contains a smaller group which has a nice property. Here is one such result of Ramsey Theory, which has a truly brilliant, elementary proof:
Statement: Suppose within a group of 6 people, every pair of people either shake hands or high-five, then at least one of the following is true: a subset of 3 people can be taken out, all of whom have shaken hands, or a subset of 3 can be taken out such that all of whom have high-fived.
Proof:
Let's label our people A, B, C, D, E, and F. Then there are 5 other people besides A. So either A has shaken hands with at least three of them or A has high-fived a group of at least three of them (it can't be both, if he's high-fived 3 and shaken hands with 3, that makes six, more than the total of 5 people other than A).
Let's say then, for example, it so happens that he has shaken hands with B, C, and D (we can relabel the people and rename a high-five if needed, such that this is true, in a process called WLOG, or without loss of generality).
Then either:
- One pair of these three people has shaken hands, in which case we're done (say B and C shook hands, then A, B, and C have all shaken hands).
- Or no pair has, in which case B, C, and D are a group of three such that none have shaken hands.
Q.E.D. (this means you have completed the proof!)
💡 Things to consider
Now let's generalize: How many people would we need such that it is necessarily true that there are two different cliques (i.e. a group of m people and n people all of whom agree on their particular greeting method)? We write this number R(m,n). For example, it can be further shown that R(3,3)=6. That is, in any group of 6, not only have 3 people all shook hands, but a group of three people have all high-fived.
Paul Erdos: Now, in general, as m and n get bigger, the number of cases to check becomes too large. Paul Erdos, a brilliant Hungarian mathematician, supposedly once said something along the lines of: imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose instead that they ask for R(6, 6). In that case, we should attempt to destroy the aliens. He's right too. We know that R(6, 6) is between 102 and 165, but to check the number 102, for example, we must clamber through 101550 possibilities!
So what's the breakthrough?: Well, although we can't know R(n,n), even for small numbers, we do know some upper bounds. In particular, 4n has been known for a while. However, recently a breakthrough occurred. In 2023, Sahasrabudhe, Morris, Griffiths, and Campos found a new upper bound. What is it? 3.9921875n. Can you believe it? Nor can I! It has revolutionised the game! I wonder what these mathematician-types will uncover next!
🔎 Find out more
🍒 The cherry on top
🎮 Can Video Games Improve Learning?: Games like Minecraft are increasingly used in classrooms. But do they actually help students learn better? This blog is a great read for those interested in Psychology, Education or Computer Science.
🌍 Global Tipping Points: Scientists warn that parts of Earth’s climate system may be approaching irreversible tipping points. But how close are we really? This website breaks down the risks and what it means for our future, and is ideal for Geography or Environmental Science students.
👽 Are We Alone in the Universe?: Scientists continue to search for extra-terrestrial life, but what evidence do we have so far? Or are we alone in the universe? This article is well worth a read for those interested in Astronomy or Physics.
👀 Keep your eyes peeled for…
Tuesday 31st March
Thursday 2nd April
🗳️ Poll
How was today's email?
That’s it for this week! We’d like to thank this week’s writer: Ben Watkins.
💚 Like UniScoops?
Forward this edition to someone who’d love to read it for extra kudos!
📢 Want to tell us something?
Reply to this email to tell us what you think about UniScoops, or to give us any suggestions on what you’d like to see.
🧐 New to UniScoops?
Get your weekly fix of academia with our fun, thought-provoking newsletter. No jargon, no fluff, just the good stuff. Subscribe today.
