current view
OpenAI construction
point
Square grid link
OpenAI construction link
The Erdős unit distance problem asks a child-simple, mathematician-hard question: if you place n points on a flat plane, how many pairs can be exactly one unit apart? The Square grid gives the natural baseline: almost 2n unit links. The OpenAI construction proves that this baseline is not the right long-run intuition: for infinitely many huge values of n, there are point sets with at least n1+δ unit links for a fixed δ > 0.
Use algebraic number theory to create many independent choices that act like hidden switches.
A class-group argument keeps many choices that can be combined into the same kind of object.
Each compatible choice creates a move whose length is exactly 1 in every relevant complex view.
Put many lattice points inside a high-dimensional rounded box so many point-plus-move pairs remain inside.
Look at one complex coordinate. Each surviving hidden move becomes a one-unit segment in the plane.
The accessible overview of the result and why it matters.
The proof document with the theorem and construction.
A digested, human-verified discussion of the counterexample and its mathematical context.
A short reference definition of the problem.
Problem-history page with known bounds and Erdős’s prize framing.
Terence Tao’s optimization-problems page summarizing exponent bounds.
A readable news explanation of the breakthrough.
Mathematician commentary and links around the result.