Point set
Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.
The line with only two points on it
Scatter finitely many points on a page, not all in one line, and draw every line through two or more of them. However cunningly the points are placed, some line ends up carrying exactly two — and the proof is a minimisation with no algebra in it at all.
Three ordinary lines from a count
Kelly's proof finds one line through exactly two of the points by minimising a distance. Melchior, seven years earlier, had found three — by turning every point into a line and counting the corners, edges and regions of the picture that results. Euler's formula for the projective plane does the rest, and it says exactly which configurations have no more than three.
The fewest ordinary lines a polygon allows
Take the corners of a regular polygon and add the points at infinity where its parallel chords meet. Every chord then carries three points, the line at infinity carries all the new ones, and the only lines left with exactly two points are the tangents at the corners — half as many as there are points. Dirac guessed in 1951 that nothing does better, and Green and Tao proved it in 2013.
At least as many lines as points
Sylvester's theorem says some line through two of the points misses all the rest. Remove one end of that line and the line itself disappears, taking at least one line away with one point. Run that backwards and it proves that n points not all in a line determine at least n lines — and the only sets that manage exactly n are a line of n − 1 points with one point off it.
Rows of three, planted on a cubic
Nine trees can be planted in ten rows of three, and the arrangement that does it is not a grid or a star but nine points on a cubic curve. On the curve three points are in line exactly when their angles add up to a right angle, so choosing the points as a cyclic group turns collinearity into addition — and the count of rows it produces is the number Green and Tao proved is the most any planting can reach.
The busiest point in a set
Scatter points in the plane, not all on one line, and draw every line through two of them. Some point lies on many of those lines — but how many must? Gabriel Dirac guessed half: in any such set, some point is on about half of the lines it could be on. Split the points evenly between two lines and no point does better than half plus one, so the guess cannot be raised. Proving it is another matter. Seventy years on, the best theorem says a third.
Every two edges must meet
A planar drawing has edges that avoid each other. Turn the rule round and insist that every two edges meet exactly once, at a shared end or a single crossing, and the drawing is what John Conway called a thrackle. In the 1960s he conjectured that a thrackle can never have more edges than points, and put up a prize. With straight edges a short argument settles it, and the argument also bounds the longest distances in a set of points. With curved edges it is still open.
How often points can sit on lines
A million points and a million lines: how many times can a point lie on a line? Counting pairs says about a billion, and over a finite field that is exactly right. In the real plane the true limit is about a hundred million, because a line has an order along it. The proof takes four lines and uses nothing but a crossing count, and from it follows the fact that no set of numbers can have both few sums and few products.
Named alongside it
The objects these essays reach for when they reach for this one.
IncidenceExtremal configurationOrdinary lineProjective planeConjectureExhaustive searchCounting argumentCrossing numberExtremal problemLower boundPlanar graphCollinear