Combinatorics

Imagine you scatter a bunch of dots on a piece of paper. The question is: how many pairs of those dots can be exactly 1 inch apart? The Erdős unit distance conjecture asked whether there's a specific mathematical formula that limits how often this can happen as you add more and more dots. Think of it like asking how many friendships can exist in a town where friends are defined as people who live exactly one mile apart — there's a suspected maximum, and Erdős guessed what that maximum should be. For decades, no one could prove or disprove his guess. Now, an AI apparently found a specific arrangement of dots (a 'counterexample') that breaks the expected limit, proving Erdős's conjecture was wrong. A team of elite mathematicians then checked and explained the AI's work in this paper.