Take a set of equiangular lines. Associate to each line a vector — an object that points in a particular direction. For each line you can imagine two possible vectors, one pointing one way on the line, the other pointing the other way. Choose one, then take the “dot product” of each pair of vectors. If the vectors form an acute angle, the dot product will be positive. If they form an obtuse angle, it will be negative.

The major innovations in the new work appeared after the authors recast the problem in the language of graph theory. Graph theory is the study of how points can be connected to each other by edges. In this scenario, the points of the graph represent the vectors. Points are connected to each other according to this rule: Color the edge between them red if the dot product is positive, blue if the dot product is negative. The result will be a configuration of red and blue lines that provides a different way of looking at the original situation.

“This is another way of encoding the information you’re given, but it’s quite suggestive,” said Keevash. “Once you have a graph, combinatorial ideas come into play.”

In particular, the authors use something called Ramsey’s theorem to bring order to the way all the red and blue edges are joined together. Ramsey’s theorem says that a graph of this sort will always contain large subsets of a certain minimum size that are completely uniform — either all red or all blue. In the case at hand, we know it’s impossible to have many vectors pointing in opposite directions, so the dominant subset of lines will always be red, not blue.

This large subset of red edges forms what Sudakov calls an “anchor” from which he and his collaborators are able to fill out the rest of the graph. By manipulating the remaining vectors, they prove that almost all the vectors not in the subset are joined to the subset via red edges. This, in a sense, casts the blue edges to the outskirts of the graph, and gives the authors a complete, well-ordered picture of what a graphical representation of a set of equiangular lines would look like — if those lines were to exist.

The authors took this arrangement of vectors and further simplified the picture by “projecting” it down into lower dimensions, where additional aspects of their structure came into view.

“It’s a bit like taking a light and shining it on an object and looking at the shadow,” said Jonathan Jedwab, a mathematician at Simon Fraser University in British Columbia who studies equiangular lines. “If you have a three-dimensional object and literally shine a light on it, the shadow it casts on the two-dimensional plane will tell something about what’s going on. If you then moved the 3-D object and shone the light again you could compare the 2-D shadows and learn even more.”

Following the projection, the authors changed settings one final time, reinterpreting their graph as a matrix — a square table of entries. In this table, each entry is a dot product of two vectors. Mathematicians commonly use these kinds of matrices — called Gram matrices — to study configurations of vectors, and in particular those coming from equiangular lines. In this new work, though, the authors had the advantage of first having used Ramsey’s theorem to understand something about the structure of vector relationships.

“After identifying this set, suddenly we cleaned the whole picture and the matrices we got later were much more structured,” said Sudakov.

Through a variety of manipulations, the researchers calculated the “rank” of these structured matrices. Rank is a basic attribute of any matrix. It quantifies, in a sense, how much information the matrix contains, or how many rows you need in order to be able to generate all the rows. (A rough analog of rank would be to count the number of primes needed to express the prime factorization of a number — a number with a longer prime factorization would be considered more complex and have a higher “rank.”)

In this setup, the rank of the matrix is both related to the number of equiangular lines and sets a limit on the number of spatial dimensions in which the lines live. As a result, the authors were able to prove that when you begin by fixing the angle in advance, the maximum number of equiangular lines is 2d – 2 for one particular angle (approximately 70.7 degrees), and no more than 1.93d for any other angle. The authors ended up needing a roundabout process to arrive at such exact numbers, but sometimes it takes a surprising series of recollections to find your way back to last night’s dream.

“My reaction isn’t ‘Why didn’t I think of it?’ It’s ‘My goodness, what an array of tools these people have,’” said Jedwab. “To string these tools together one after another, I think that’s the real ingeniousness of what they’ve done.”