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

Abstract: We resolve a $1000 Erd\H{o}s prize problem, complete with formal verification generated by a LLM. 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.

• The paper disproves Erdős's conjecture by demonstrating that certain finite Sidon sets cannot be extended to perfect difference sets.
• It employs a human-assisted proof methodology combining LLM-generated Lean code with traditional combinatorial arguments.
• The results impact combinatorial design theory and projective geometry while advancing AI-supported formal verification.

Forbidden Sidon Subsets and Human-Assisted Formal Proofs

Introduction and Problem Statement

This paper addresses a longstanding conjecture in additive combinatorics, originally posed by Paul Erdős, concerning the relationship between Sidon sets and perfect difference sets. Specifically, Erdős conjectured that every finite Sidon set can be extended to a finite perfect difference set. The authors provide explicit counterexamples to this conjecture, resolve a \$1000 prize problem, and present a formal verification of their results using a human-assisted proof methodology involving LLMs and the Lean proof assistant.

A Sidon set $A$ is defined such that all differences $a-a'$ of distinct $a, a' \in A$ are distinct. Equivalently, all pairwise sums $a+a'$ are distinct up to reordering. A perfect difference set $B$ modulo $v$ is a subset of $\mathbb{Z}/v\mathbb{Z}$ such that the differences $b-b'$ for distinct $b, b' \in B$ cover every nonzero residue exactly once. Singer's theorem guarantees the existence of perfect difference sets for $v = q^2 + q + 1$ when $q$ is a prime power.

Disproof of Erdős's Conjecture

The main results are twofold:

• Theorem 1: The Sidon set $\{1,2,4,8\}$ cannot be extended to a perfect difference set modulo $v = p^2 + p + 1$ for any prime $p$.
• Theorem 2: The Sidon set $\{1,2,4,8,13\}$ cannot be extended to a perfect difference set modulo any $v > 0$.

These results directly contradict Erdős's conjecture, establishing that not all finite Sidon sets can be embedded in a perfect difference set. The authors also note that Marshall Hall, Jr. had previously published a counterexample in 1947, which had been overlooked in the literature.

The proofs utilize both direct combinatorial arguments and geometric constructions involving finite projective planes and polarities. The projective plane approach leverages the correspondence between perfect difference sets and cyclic projective planes, and applies results from Baer on polarities and absolute points to derive necessary conditions that are violated by the counterexamples.

Formal Verification via Human-Assisted Proof

A significant methodological contribution is the use of LLMs (specifically ChatGPT-5) to generate Lean code for the formalization and verification of the counterexamples. The authors describe a workflow in which the human mathematician interacts with the LLM to translate informal mathematical arguments into formal Lean proofs. The resulting formalization comprises thousands of lines of Lean code, including definitions, lemmas, and theorems, and is verified using the Mathlib library.

The process revealed several practical challenges, such as the complexity of formalizing simple combinatorial arguments (e.g., involutions and parity), and the limitations of LLMs in handling certain aspects of Lean's type system and cardinality notions. Nevertheless, the Lean proof provides a machine-checked guarantee of correctness, overcoming the unreliability of informal or LLM-generated proofs.

Implications and Theoretical Significance

The explicit construction of forbidden Sidon subsets has several implications:

• Combinatorial Design Theory: The results clarify the limitations of extending Sidon sets to perfect difference sets, impacting the construction of combinatorial designs and error-correcting codes.
• Projective Geometry: The connection to cyclic projective planes and the prime power conjecture is reinforced, with the counterexamples providing further evidence for the necessity of prime power orders in the existence of perfect difference sets.
• Formal Methods in Mathematics: The successful use of LLMs and Lean for formal verification demonstrates the feasibility and utility of human-assisted proof workflows, suggesting a paradigm shift in mathematical research and verification.

The paper also discusses the "de Bruijn factor"—the ratio of formal to informal proof length—and notes that formalization overhead varies significantly depending on the argument's nature.

Future Directions

Open questions remain regarding the minimal size $s$ of forbidden Sidon sets and the classification of all such sets. The authors suggest further exploration of historical problems that may have been solved prior to their formal posing, and speculate on the potential for AI to resolve other high-value Erdős problems. The integration of LLMs with proof assistants is identified as a promising avenue for improving the efficiency and accessibility of formal mathematical research.

Conclusion

This work provides a definitive resolution to a long-standing conjecture in additive combinatorics, establishes explicit forbidden Sidon subsets, and demonstrates the practical application of human-assisted formal proof using LLMs and Lean. The results have both combinatorial and methodological significance, and point toward a future in which AI-assisted formalization becomes an integral part of mathematical practice.