In this episode of our podcast My Favorite Theorem, I talked with Yen Duong, who has a Ph.D. in math and now works as a science writer, a little like another math and science writer you might know and love. We were not joined by my cohost Kevin Knudson because we recorded the episode in person at the annual meeting of the National Association of Science Writers. You can listen to the episode here or at kpknudson.com, where there is also a transcript.

Dr. Duong decided to talk about what she refers to as a gateway math proof, a fact about acquaintances and strangers. As she framed the question, suppose you are trying to form a committee of three people and you want all three to be acquaintances or all three to be strangers. How big does the pool of potential committee members need to be to guarantee that you can form such a committee? Three people wouldn’t work because two of them could know each other and the other could be a stranger to both of them. In fact, a pool of six will be sufficient, as Dr. Duong explains. (For a written rather than spoken explanation, check out her post about it.) The theorem is often phrased in terms of the social dynamics of parties rather than committees, and Dr. Duong likes to tell people about it at parties, so it is a cocktail party theorem on two levels.

This theorem about Ramsey numbers, sometimes called the friends and strangers theorem, is fun because it is accessible, no matter your math background, but leads very quickly into open questions in math. The minimum number of people you need to guarantee a trio of friends or a trio of strangers is called the Ramsey number R(3,3), and proving that it is six is doable with pencil and paper. The equivalent question for four mutual friends or strangers—finding the Ramsey number R(4,4), logically enough—is much more difficult but was solved in the 1950s. For five friends or strangers, the Ramsey number R(5,5) is unknown. If you’d like to learn more about Ramsey numbers and how little we know about them, check out these posts.

In each episode, we ask our guest to pair their theorem with something: food, beverage, art, music, or other delight in life. Dr. Duong chose broccoli with cheese sauce. You’ll have to listen to the episode to find out why it is the perfect accompaniment to the friends and strangers theorem.
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