I mentioned a couple weeks back reading John Stillwell’s Roads To Infinity: The Mathematics of Truth and Proof and how it stirred my desire to do mathematical logic. Besides that it reminded me of a baffling thing I’d read sometime around 1980. My memory is too vague to pin down the year nearer than that, but it was surely sometime between 1979 and 1983.

It draws on some newspaper column, I think a letter to Dear Abby or Dear Ann Landers. The letter-writer was complaining about ivory-tower academicians such as (to paraphrase) “mathematicians who work on how many people can be at a dinner party without three knowing each other instead of on solving world hunger”. The complaint struck me as unfair as a kid. The skills that make good mathematicians don’t have to have anything to do with feeding people. And it struck me even back then there was probably enough food produced. It was just not getting to hungry people for reasons that were likely, at heart, evil. (Today, I hold basically the same view.) Still the letter struck me as weird because … … Well, even granting the argument that mathematicians could be working on world hunger instead, what is Dear Abby supposed to do about it? (That I don’t remember Dear Abby’s response suggests maybe it was some other feature, or perhaps the letters to the editor. Or that she had no good answer.)

Roads To Infinity brings this old complaint back to mind because among its pages it discusses Ramsey Theory. This is a section of mathematics interested in combinatorics and graph theory. Its questions are like: how many ways can you arrange things that connect to one another with certain restrictions? And the dinner-party thing is the one piece of Ramsey Theory that any normal, non-mathematician might have heard of. This is because it’s a theorem that can be put into an immediately accessible, immediately understandable form. Even a seven-year-old can understand the question. The seven-year-old could even follow a demonstration of why the proof is true. The seven-year-old might even follow the proof, because it’s easier than you might guess.

The problem alluded to by the Dear Abby(?) complainer, and discussed in Roads To Infinity, is: what is the smallest number of people you must invite to a party to be sure that either at least three of them all know one another, or that at least three of them do not know one another? This is the simplest interesting example of the “party problem”. It asks how many things you need to gather so as to be sure that either some number m share a property, or some number n of them do not. I’ll not spoil the fun for people who want to work out this particular case.

What’s interesting about the result is that it suggests you can’t avoid structure. Get together enough things that can either have or not-have a relationship between pairs. Furthermore you will get relationships among bigger groups. We could interpret this as a reason there must be coincidences; logic compels them. The field speaks to us about how things must relate to one another.

But here’s what has me baffled: why was the Dear Abby(?) letter-writer aware of Ramsey Theory? What was going on in United States pop culture of the late-70s or early-80s that *this* might have been on the complainer’s mind? Why not something at least as abstract and more accessible, like the Goldbach conjecture? (That’s the notion any even number greater than two can be written as the sum of two primes?) Did something tell people this dinner-party problem was something mathematicians had worked on? Did Johnny Carson make a monologue joke about it?

The original problem, as best I can figure, was solved in 1930. Perhaps there was a surprising improvement in the proof that made it newsworthy at the time. I don’t know the history of mathematics in the 1970s in the right detail for that. Was it a recreational-mathematics challenge going around, the way a couple months ago everybody was worked up about that Singapore Birthday problem? Was there a good bio-pic of Frank Plumpton Ramsey that came out around that time?

I don’t know what motivated the letter-writer to start. Nor do I know why the memory of that letter should have lasted in my mind. I am curious if someone can suggest why the subject ever entered the realm of people complaining in newspapers, though.

Maybe the letter writer and a mathematician were at a dinner party together. Maybe they knew each other or they didn’t know each other and the mathematician commented on it. Maybe the dinner party was ruined because of it.

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All quite possible. I wonder if they knew each other before the dinner party.

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n = 4 is not it. If you have four people where only A knows B, B knows C, and C knows D, the requirements are met.

I suspect n = 5 is the solution because I can’t come up with a counterexample, but I lack a proof.

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Bah, I have failed to recognize a trivial case, n = 5 is not correct either.

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Yeah, 5 is just short of doing it. But you’re getting there. If you get to n = 260 you’ve gone too far.

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If I recall the story correctly, there was a newspaper article about the Ramsey theory discovery. They were simply responding to that.

The weird thing is, if any kind of mathematics is going to help solve world hunger, it is graph theory, which is enormously helpful in logistics.

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Ah, now, this is interesting and maybe I can find some leads in newspaper archives. Thank you.

And yeah, it is a weird complaint. For all the beautiful abstractness of graph theory, it also has this immediate application to anything that networks. The dinner-party example is a good case of that. There’s whole fields of mathematics that can’t be made as immediately understandable as that.

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Hey Joseph – I’ve got some (possibly?) interesting info about this for you, but it’s rather long, and the email address I’ve successfully used for you before is giving me server-lever “does not exist” errors. If/when you see this (and if you are interested!) shoot me an email from an address you prefer and I’ll send the info along. Happy Thanksgiving – Tony

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Oh, my, thank you. Shall send an e-mail shortly. Thanks and I hope yours was a happy Thanksgiving.

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