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## Thursday, November 29, 2018

### Graph Theory and Cocktail Parties | Roots of Unity - Scientific American

Science writer Yen Duong tells us why she loves the "friends and strangers" theorem, says Evelyn Lamb, Freelance math and science writer based in Salt Lake City, Utah.

 Social connections are rich fodder for the field of graph theory

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.

 Yen Duong. Photo: Yen Duong
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...

You can find Dr. Duong on her website, her blog Baking and Math, and Twitter. Some of her writing on science and health appears at North Carolina Health News. As we mention in the episode, both she and I started our science writing careers through the AAAS Mass Media Fellowship program. This paid internship places math and science graduate students or recent graduates in news rooms around the country for a summer. Our fellowships were sponsored by the American Mathematical Society. Both of us consider it formative in the direction our careers have taken. You can read about her experience with the Mass Media Fellowship in the Notices of the American Mathematical Society.