Terence Tao @tao@mathstodon.xyz
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Inspired by the recent successful "vibe coding" in Lean of a solution to an Erdos problem at arxiv.org/abs/2510.19804 , I tried my hand at formalizing the disproof (by Pikhurko in 2001) that had recently been unearthed by Quanyu Tang of another Erdos problem erdosproblems.com/613. This was a simple finite counterexample: a graph of 44 edges on 15 vertices that could not be split into a bipartite graph and a graph with max degree 5. The paper of Pikhurko is just 9 pages (and also constructs more general counterexamples with larger choices of parameters, as well as some reasonably matching lower bounds), so this looked like within reach of modern AI tools. I therefore uploaded the paper to ChatGPT Pro and asked it first to summarize the construction in informal language, and then formalize step by step. chatgpt.com/share/690a65b3-473 (1/4)

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

We resolve a $1000 Erdős 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 Erdős 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 Erdős 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.

arxiv.org · arXiv.org

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