The proof of the Harary--Hill conjecture: a detailed exposition and an extension to rotation systems

Abstract

The Harary--Hill conjecture states that the crossing number $\ucr{K_n}$ of the complete graph $K_n$ is $Z(n):=\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor$ for all $n\ge 3$. In October 2026 OpenAI uploaded to GitHub a proof of this conjecture. The proof is very short, but it relies on polynomial methods that are not part of the standard techniques used by many combinatorialists, and it is written with very little motivation or explanation. Since we found the proof very hard to follow, we decided to write this paper, which contains a complete proof of the conjecture written for mathematicians who are more familiar with combinatorics than with polynomial methods. In fact we prove a more general result on rotation systems of $K_n$, from which the Harary--Hill conjecture easily follows. Namely, we show that for every rotation system $Π$ of $K_n$ the number of $4$-vertex subsystems of $Π$ realizable as drawings of $K_4$ with one crossing, plus three times the number of non-realizable $4$-vertex subsystems of $Π$, is at least $Z(n)$. The statement and proof of our result are purely combinatorial and do not involve drawings at all.