Consider an n-dimensional hypercube, and connect each pair of vertices to obtain a complete graph on 2n vertices. Then colour each of the edges of this graph using only the colours red and black. What is the smallest value of n for which every possible such colouring must necessarily contain a single-coloured complete sub-graph with 4 vertices which lie in a plane?
I watched the Horizon programme last night on infinity, which began very promisingly when it was on the subject of large numbers and mathematics, but then took a speculative turn for the worse, when it began to make all kind of assertions about an infinite universe, infinite parallel universes, identical versions of people, and so on, without giving anything really in the way of hard science.
The mathematics was pretty good. After a brief digression with so-called "large numbers", which involved a googolplex and Graham's number (which was only vaguely stated as a boundary solution to a problem), we were treated to a good illustration of Hillbert's paradoxical "infinite hotel", and what it showed about infinity - and the dangers of just treating infinity like any other number, although it stopped there, just on the verge of the branch of mathematics known as analysis was; it might have mentioned that kind of mathematics was developed for avoid paradoxes with infinity. And as might have been expected, we also had Cantor on infinity defined by counting by matching one set against another, and different infinities, which are now part of any mathematical syllabus.
A set of elements can be defined as infinite if the set has a seemingly paradoxical quality: a subset of elements in an infinite set can be matched up, one-to-one, to all of the elements in a set
It would have been nice to hear about the continuum hypothesis (which after all is related to different infinities), and back in the 1980s, I remember a Horizon on the unsolved problems of mathematics which did just that. But now we are 2010, and programmes are dumbed down to what a character in one of Gregory Benford's novels called a public 'with the attention span of a commercial'.
Professor Gregory Benford's book Cosm, which is a good adventure story with cutting edge physics does explore exactly what parallel universes might be like, and how the physics of this works. This is the publisher's blurb, which gives some idea of the story.
Of all the things one expects to get from slapping heavy nuclei together in a high-energy accelerator one would not expect a football-sized particle, yet that is exactly the result Prof Alica Butterworth got when she started hurling uranium nuclei together at close to light speeds: well, that and a wrecked collider. Amid the wreckage her colleagues do not realise what the 'football' may be, so she was able to discretely remove it to her own university lab. There she finds that the object may hold the secret to the Universe's creation. However physicists from the collider are hot on her trail. Who will unravel the mystery first?
Benford uses hard edge cutting physics; he is an active professor of physics himself. But instead of that, there was a lot of talk, mostly made interesting (or so the programme makers must have thought) by bizarre lighting, a sinister face (Steven Berkoff ) looming in black and white, and intoning in a sepulchral voice (suitably jazzed up by some kind of voice modulator). I don't know who puts these programmes together, but they seem to assume people have the attention span of a pickled newt.
Yet Armand Leroi on Channel 4 ("Aristotle's Lagoon") or BBC4's "Chemistry a Volatile History" show that science can be presented without dumbing down. But instead of a physics of Benford calibre, we got a someone telling us that ""I think the universe is infinite on Mondays, Wednesdays and Fridays, and I think it's finite the rest of the week. I'm having a really hard time making my mind up."
There were periodic diversions to a monkey in front of a typewriter, supposedly typing what people had just said, while a voiceover repeated it. This was an allusion to the statistical paradox - if infinitely many monkeys are set before typewriters, with infinite lifespans and infinite time, the statistical paradox goes, they will sooner or later produce Shakespeare's plays.
But the programme, while starting well, lost its way when it began to explore the physics without any physics, and I'd sooner recommend Benford's Cosm to anyone who really wants to explore the infinite.
'". Well, the Cosm isn't just a footprint it's the real thing, a direct quantum gravity artifact sitting still in the lab, big enough to put your hands on. We're taking fundamental physics back to a human scale!"' (Gregory Benford, Cosm p253)
'To peer through the quick stubble of mathematics and see the wonders lurking behind was to momentarily live in the infinite, beyond the press of the ordinary world where everyone else dwelled in ignorance.' (Cosm, p333 )
Links:
http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel
http://www.amazon.co.uk/COSM-Gregory-Benford/dp/1857236270/ref=sr_1_1?ie=UTF8&s=books&qid=1265895945&sr=1-1
http://en.wikipedia.org/wiki/Graham's_number
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