The Ramsey number R(m,n) is the minimum size of a fully connected graph with edges colored red or blue such that there is either an m-sized blue clique or an n-sized red clique. R(3,3) is 6, which is stated as every group of 6 having 3 mutual friends or 3 mutual strangers. Ramsey showed that these exist for every m and n, I think.
But they grow very rapidly. Ramsey theory has usually the biggest constants found in mathematical papers.[1] There is a famous quote by Erdos:
"Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack."
What always gets me about Graham's Number is that you can't even write down how many digits it has using the entire observable universe ... but "the last 12 digits are ...262464195387".
Add Turing to the list, having him around during the naissance of molecular genetics and the flowering of computation would have made those fields all the more lively.
Euler is really the master in terms of sheer volume and mathematical scope. You supposedly could get a theorem named after you if you discovered it in Euler’s notebooks (though probably well picked over by now).
Not that well picked over. After about 100 years of work and several volumes being published, the work on producing the complete collected works of Euler is not done.
> Lydia made the mistake of remarking, “What a beautiful tree,” presumably too casually, whereupon Wittgenstein glared and demanded, “What do you mean?” and she burst into tears.
There’s lots of great stuff in Ramsey theory. In some sense it feels quite different to a lot of other mathematics. It also has the great property that it is typically very hard to look at a statement in Ramsey theory and know if it is likely true or not, or even if it will likely be easy to prove or not.
For example consider a recently disproven long-standing problem of Ramsey theory:
Is it possible to colour the natural numbers with two colours such that no Pythagorean triple is all one colour?
This question was asked by Ron Graham many years ago with a $100 prize for the answer, assuming it would be reasonably easy to prove. It was recently solved with an absolutely enormous multi-terabyte computer proof. It turns out you can colour everything up to 7824 fine but no choice for 7825 is sufficient.
If anyone from the New Yorker sees this, here is what your user interface looks like to the average reader of your online newspaper: https://imgur.com/a/kCoxLHR
Given the media industry's history, I'm sure if anyone relevant does see your post, that white space is simply going to be used as display ad real estate.
>>Discounting the interests of future people, Ramsey wrote, is “ethically indefensible and arises merely from the weakness of the imagination.
Climate change, the federal debt, increasing shareholder value over anything else, I wonder what Ramsey would have to say about the current state of the world.
He might well say that the problem is that it is not growing fast enough and that concerns about the federal debt or climate change are morally indefensible. (An example of this: the coronavirus crisis has exposed the scleroticism and poverty of the present, and the extent to which the 'unseen' has been sacrificed by regulators and how other less important issues have distracted from important things like pandemic prevention.)
If you take Ramsey's ideas about intertemporal consumption and the bliss point seriously and you try to do an empirical calculation, it generally implies that the utilitarian thing is to have savings/investment rates anywhere up to 98% (!!!) vs the current savings rate of ~0%: https://plato.stanford.edu/entries/ramsey-economics/#NumeEst... See also 'turnpike theory' which builds on Ramsey and also finds that generally, the optimal thing is to grow as fast as possible to hit the limits as quickly as possible and only then start consuming: https://en.wikipedia.org/wiki/Turnpike_theory
Climate change is bad but the federal debt is good. It allows the private sector to be net savers and creates overall stability. It’s also clearly sustainable, hundreds of years and counting.
If there's one thing sorely lacking from the public discussion about what you mentioned, it's a balanced and thorough evaluation including reliable estimates for the effects.
The federal debt has had no observable effect whatsoever for decades, the dollar is as strong and internationally valued as ever. Climate change has not made anything measurably worse in the 3 decades people have been screaming about it. The bit about shareholder value is too unspecific.
> (As an Oxford mathematician, Martin Gould, has explained, Ramsey theory tells us, for instance, that among any six users of Facebook there will always be either a trio of mutual friends or a trio in which none are friends.)
What is the sensible/true version of this? Clearly it’s not correct as stated. You could have three pairs of two mutuals. You could have five mutuals and a singlet. Or six mutuals. Is it more a statement about probabilities?
Let the 6 nodes be users, and color edges between two users red if they are friends, blue if they are not. I think it's true as stated, as another comment claims.
[+] [-] ur-whale|5 years ago|reply
[+] [-] sn41|5 years ago|reply
But they grow very rapidly. Ramsey theory has usually the biggest constants found in mathematical papers.[1] There is a famous quote by Erdos:
"Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world's best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack."
[1]https://en.m.wikipedia.org/wiki/Graham%27s_number
[+] [-] zimpenfish|5 years ago|reply
I swear mathematicians are witches.
[+] [-] schr0dinger|5 years ago|reply
https://en.m.wikipedia.org/wiki/Aleph_number
[+] [-] brianyu8|5 years ago|reply
[+] [-] pharke|5 years ago|reply
[+] [-] markdog12|5 years ago|reply
[+] [-] smitty1110|5 years ago|reply
[+] [-] schintan|5 years ago|reply
[+] [-] gumby|5 years ago|reply
[+] [-] dan-robertson|5 years ago|reply
[+] [-] hetspookjee|5 years ago|reply
[+] [-] papeda|5 years ago|reply
> Lydia made the mistake of remarking, “What a beautiful tree,” presumably too casually, whereupon Wittgenstein glared and demanded, “What do you mean?” and she burst into tears.
[+] [-] dan-robertson|5 years ago|reply
For example consider a recently disproven long-standing problem of Ramsey theory:
Is it possible to colour the natural numbers with two colours such that no Pythagorean triple is all one colour?
This question was asked by Ron Graham many years ago with a $100 prize for the answer, assuming it would be reasonably easy to prove. It was recently solved with an absolutely enormous multi-terabyte computer proof. It turns out you can colour everything up to 7824 fine but no choice for 7825 is sufficient.
[+] [-] pgt|5 years ago|reply
Notice the % that is content.
[+] [-] jahn716|5 years ago|reply
[+] [-] rv-de|5 years ago|reply
Wasn't he a pretty ruthless and competitive guy?
[+] [-] frankbreetz|5 years ago|reply
Climate change, the federal debt, increasing shareholder value over anything else, I wonder what Ramsey would have to say about the current state of the world.
[+] [-] gwern|5 years ago|reply
If you take Ramsey's ideas about intertemporal consumption and the bliss point seriously and you try to do an empirical calculation, it generally implies that the utilitarian thing is to have savings/investment rates anywhere up to 98% (!!!) vs the current savings rate of ~0%: https://plato.stanford.edu/entries/ramsey-economics/#NumeEst... See also 'turnpike theory' which builds on Ramsey and also finds that generally, the optimal thing is to grow as fast as possible to hit the limits as quickly as possible and only then start consuming: https://en.wikipedia.org/wiki/Turnpike_theory
[+] [-] slivanes|5 years ago|reply
[+] [-] viburnum|5 years ago|reply
[+] [-] lazyjones|5 years ago|reply
The federal debt has had no observable effect whatsoever for decades, the dollar is as strong and internationally valued as ever. Climate change has not made anything measurably worse in the 3 decades people have been screaming about it. The bit about shareholder value is too unspecific.
[+] [-] mordymoop|5 years ago|reply
What is the sensible/true version of this? Clearly it’s not correct as stated. You could have three pairs of two mutuals. You could have five mutuals and a singlet. Or six mutuals. Is it more a statement about probabilities?
[+] [-] MaysonL|5 years ago|reply
[+] [-] wging|5 years ago|reply
Let the 6 nodes be users, and color edges between two users red if they are friends, blue if they are not. I think it's true as stated, as another comment claims.