If you have lots of objects lined up in a row, and only a relatively small palette of colours to paint them with, then you’ll expect to be able to find some patterns lurking in any colouring of the objects.

Here’s a famous and lovely combinatorial theorem to that effect due to Van der Waerden.

For any* r *and* k*, there is an *N* big enough so that, however the numbers 1, 2, 3, … *N* are coloured with *r* colours, there will be an monochromatic arithmetic progression of numbers which is *k* long.

For example, suppose you have *r* = 7 colours, and put *k* = 4, then there is a number *N* = *W*(*7*, 4) such that, it doesn’t matter how you colour the first *N *or more* *positive integers with 7 colours, you’ll find an arithmetical sequence of numbers *a*, *a + e*, *a* + 2*e*, *a* + 3*e * which are all the same colour. As is so often the way with numbers that crop up in this sort of combinatorics, no one knows how big *W*(*7*, 4) is: the best published upper limit for such numbers is huge.

Here’s a simple corollary of Van der Waerden’s theorem (take the case where *k* = 4, and remark that *a +* *a* + 3*e = **a + e +* *a* + 2*e*)

For any finite number of colours, however the positive integers are coloured with those colours, there will be distinct numbers *a*, *b*, *c*, *d* the same colour such that *a + d = b + c.*

So far so good. But now let’s ask: does this still hold if instead of considering a finite colouring of the countably many positive integers we consider a countable colouring of the uncountably many reals? In other words, does the following claim hold:

(E) For any \(\aleph_0\)-colouring of the real numbers, there exist distinct numbers *a*, *b*, *c*, *d* the same colour such that *a + d = b + c.*

Or since a colouring is a function from objects to colours (or numbers labelling colours) we can drop the metaphor and rephrase (E) like this.

(E*) For any function \(f : \mathbb{R} \to \mathbb{N}\) there are four distinct reals, *a*, *b*, *c*, *d* such that *f*(*a*) = *f*(*b*) = *f*(*c*) = *f*(*d*), and *a + d = b + c*.

So is (E*) true? Which, you might think, seems a natural enough question to ask if you like combinatorial results and like thinking about what results carry over from finite/countable cases to non-countable cases. And the question looks humdrum enough to have a determinate answer. No?

*Yet Erdős showed that (E*) can’t be proved or disproved by ZFC. *Why so? Because (E*) turns out to be equivalent to the negation of the Continuum Hypothesis. Which is surely a surprise. At any rate, (E*) is the most seemingly humdrum proposition I’ve come across, a proposition not-obviously-about-the-size-of-sets, that is independent of our favourite foundational theory.

Make of that what you will! — but I just thought it was fun to spread the word. You’ll learn more, and be able to follow up references, in an arXiv paper by Stephen Fenner and William Gasarch here.