Where can I find a reference to the result that if the plane is colored in 4 colors, so that the set of colors of the same kind is measurable for each color, then there is a set containing 2 of distance 1?
The result was proved by Kenneth J. Falconer. The reference is
MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189.
The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue measure, and the proof makes use of a theorem of D. G. Larman and C. A. Rogers. The write up is fairly dense, though, and filling in all the details takes some work.
A more recent presentation of the proof can be found in
MR2458293 (2010a:05005). Soifer, Alexander. The mathematical coloring book. Mathematics of coloring and the colorful life of its creators. With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau. Springer, New York, 2009. xxx+607 pp. ISBN: 978-0-387-74640-1.
If you are not familiar with Soifer's book, I recommend it. It is uniquely idiosyncratic, but charming and full of interesting mathematics. Falconer's theorem is discussed in Chapter 9, "Measurable chromatic number of the plane", pp. 60–66. He writes: "I found his 1981 publication [Fal1] to be too concise and not self-contained for the result that I viewed as very important. Accordingly, I asked Kenneth Falconer, currently a professor and dean at the University of St. Andrews in Scotland, for a more detailed and self-contained exposition. In February 2005, I received Kenneth's manuscript, hand-written especially for this book, which I am delighted to share with you."
In a remarkable recent development that supersedes Falconer's result, Aubrey de Grey has found a finite set of points in $\mathbb R^2$ with chromatic number 5, meaning that if a color is assigned to each point, and only 4 colors are used, then there are two points of the same color at distance 1 from each other. De Grey's graph has 1581 vertices.
ArXiv:1804.02385. The chromatic number of the plane is at least 5. Aubrey D.N.J. de Grey, preprint.
This number has since been improved, see the page for Polymath 16 here, in particular the table in section 2.