RheumaView Insights · Mathematics
When neighbours turn out to be near-twins
A new result about Ramsey numbers does not compute a single one of them. It proves something stranger and, in its way, more useful: that consecutive members of this notoriously wild sequence grow smoothly, with no sudden lurches.
Walkthrough of the proof, step by step. An educational video produced by RheumaView™ — RheumaView did not prove this result; the video only explains a known theorem.
Start with a party. Among any six people you will always find three who all know one another, or three who are all strangers to one another. Six is the threshold. With five guests you can still arrange the acquaintances so that neither trio appears; with six you cannot, no matter how you try.
That number — six — is the simplest Ramsey number. Written in full it is R(3, 3) = 6. The general object R(k, ℓ) is the smallest crowd in which you are guaranteed either a group of k people who all know each other, or a group of ℓ people among whom there is not a single acquaintance. The party puzzle is just the small corner of a very large and very stubborn problem.
Why these numbers are maddening
You would think the natural move is to keep counting. R(4, 4) turns out to be 18. And R(5, 5)? Nobody knows. We have pinned it only to somewhere between 43 and 48, and there it has sat for years. The trouble is not laziness. The number of possible arrangements grows so fast that brute force defeats even the largest computers long before the answer comes into view.
Erdős, who spent much of his life on these questions, put it in a story he liked to tell. Suppose an alien fleet demanded the exact value of R(5, 5) on pain of destroying the planet. We should, he said, marshal every mathematician and every machine and try to find it. But if they asked instead for R(6, 6), we would do better to spend our effort building defences and preparing to fight.
So the exact values are mostly beyond reach. Mathematicians retreated to a softer question. Fix k — say, always hunt for a clique of four mutual acquaintances — and let ℓ climb. How fast does R(4, ℓ) grow? Like a square of ℓ? A cube? This is what is meant by the asymptotics: the behaviour in the limit, when the numbers get large. Even here the gaps are enormous. For most values of k the exact rate of growth is unknown.
What was actually proved
The new theorem computes no Ramsey number and settles no growth rate. It answers a finer question, one that turns out to be surprisingly delicate: how much does each number differ from the one beside it?
Fix k, and look at the ratio of two consecutive numbers,
R(k, ℓ+1) / R(k, ℓ).
As ℓ grows, that ratio tends to one. In plain terms: consecutive Ramsey numbers become almost indistinguishable in relative size. Moving from a search for ℓ strangers to a search for ℓ+1 of them changes the answer by a vanishingly small fraction.
It is worth being careful here, because the claim is easy to misread. This does not say R(k, ℓ+1) and R(k, ℓ) are close as plain numbers — the absolute gap between them can be huge. It says the gap is small compared to the numbers themselves. If a number is around a million, its neighbour might sit a thousand away: a lot in absolute terms, a sliver in proportion. The ratio comes out near 1.001, and the further along you go, the closer to a flat one it gets.
Why this was not obvious
A fair objection: surely it was clear all along that neighbours are close? We only nudged ℓ up by one.
It was not clear, and the reason is the character of these numbers. Ramsey numbers are known for behaving erratically, in fits and jumps. There was no prior reason to rule out a sudden leap somewhere in the sequence — some R(k, ℓ+1) abruptly half again as large as its predecessor. Proving that such leaps never happen, while not knowing the numbers themselves, is a different thing entirely from waving the matter away.
The proof’s cleverness is in how it routes around our ignorance. Argue by contradiction. Suppose a leap did occur, and R(k, ℓ+1) really were much larger than R(k, ℓ). One can then show that the critical graph — the extremal arrangement of acquaintances sitting right at the threshold — would have to be dense: every person knowing a great many others. Yet everything we understand about Ramsey numbers points the other way, toward arrangements that are sparse, thin on acquaintances. A leap demands density; the nature of the problem demands sparsity. That tension is enough to kill the leap. The tool used to extract the contradiction is called dependent random choice — a way of reaching into a dense graph and pulling out a well-organised piece.
The proof never needs to know the Ramsey numbers. It is rather like showing two houses on a street are nearly the same height without measuring either, judging only from how their shadows fall.
It works from the crude upper and lower bounds that have been known for decades and squeezes a sharp statement about neighbours out of them. No exact value enters anywhere.
Why bother
One might still ask what the point is, if we do not learn the numbers themselves. The point is that results like this are reconnaissance. Ramsey numbers have resisted frontal assault for close to a century. Each time something new and dependable can be proved about them — even a property like smoothness of growth, rather than a value — it is a fresh handhold on a sheer wall. Erdős asked this very question about the ratio of neighbours because he suspected the answer would say something structural about how the numbers are built.
And the method matters as much as the conclusion. Dependent random choice was devised for other purposes; that it works here, turning a hypothetical leap into an impossible density, widens the toolkit. Proofs of this kind are often a rehearsal for deeper ones. A technique sharpened on the “easy” question about neighbours can be the key, later, to the growth rate itself.
There is also a plain satisfaction in it. We have a clean, final answer to a specific question about an object that otherwise remains one of the great unsolved puzzles. We still cannot tell you R(5, 5). But we now know, firmly and provably, that these elusive numbers — wild as they are — climb without sudden jerks. A small certainty in the middle of a large unknown, of the sort that mathematics eventually builds the big certainties out of.
Olga Goodman, MD — RheumaView Insights