Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof

TL;DR

This paper solves a long-standing Erdős problem by providing a counterexample to the conjecture that every finite Sidon set extends to a perfect difference set, using human and AI-assisted formal verification.

Contribution

It presents the first known counterexample to Erdős's conjecture and employs large language models for formal proof verification, highlighting AI's role in mathematical research.

Findings

Identified {1, 2, 4, 8, 13} as a counterexample

Discovered a prior counterexample by Marshall Hall, Jr.

Used AI-assisted proof verification with ChatGPT and Lean

Abstract

We resolve a $1000 Erd\H{o}s prize problem, complete with formal verification generated by a large language model. In over a dozen papers, beginning in 1976 and spanning two decades, Paul Erd\H{o}s repeatedly posed one of his "favourite" conjectures: every finite Sidon set can be extended to a finite perfect difference set. We establish that {1, 2, 4, 8, 13} is a counterexample to this conjecture. During the preparation of this paper, we discovered that although this problem was presumed to be open for half a century, Marshall Hall, Jr. published a different counterexample three decades before Erd\H{o}s first posed the problem. With a healthy skepticism of this apparent oversight, and out of an abundance of caution, we used ChatGPT to vibe code a Lean proof of both Hall's and our counterexamples.