Paul Erdős owned almost nothing and proved almost everything. For six decades he lived out of a single suitcase, moving from one mathematician's spare room to the next, announcing at the door that "my brain is open." He wrote around fifteen hundred papers with more than five hundred co-authors — a body of work so collaborative it produced its own unit of distance, the Erdős number: your degree of separation, through joint papers, from the man himself.
His mathematics lived in combinatorics, number theory, set theory and probability. He was a founder of the probabilistic method — proving an object exists by showing a random one works with positive probability — and of Ramsey theory's central questions about the unavoidable order inside any large enough structure. He posed problems the way other people breathe, often attaching small cash prizes; some of those problems are still open, and some, eighty years on, are only now falling.
Erdős spoke of The Book: a transfinite volume, held by a deity he otherwise professed not to believe in (the "Supreme Fascist"), in which the most perfect proof of every theorem is written. You did not have to believe in God, he said, but you had to believe in The Book. To call a proof "straight from The Book" was his highest praise — not that it was correct, but that it was inevitable, the argument stripped to the one line the theorem always wanted.
Every scientist in this gallery is read as one traversal of the generative chain. Erdős is the chain run socially:
The collaboration graph is the operator chain externalised: no single mind holds the whole traversal; the network does.
In 2026 something new attached to that graph. Several of Erdős's own problems were resolved with artificial intelligence — and one of them, Erdős Problem #728, came back not as prose but as a proof written in Lean and checked, line by line, by a machine kernel. The dream of The Book met a book a computer can actually read. Erdős, who valued a proof by who could see its inevitability, would have recognised the standard, even if the reader was no longer flesh.
In 1946 he asked how often the same distance can occur among n points in the plane. He could show a square grid achieves n1+Ω(1/log log n) unit distances, and that no configuration exceeds O(n3/2) — two unit circles meet in at most two points, so the unit-distance graph contains no K2,3. Spencer, Szemerédi and Trotter brought the ceiling to O(n4/3) in 1984, where it still sits. Erdős believed the grid was essentially right — that the truth is n1+o(1) — and put money on it: $300 in 1982, $500 by 1995. It is Problem #90, and by most accounts the best-known question in discrete geometry.
In May 2026 an internal model at OpenAI produced a counterexample. There is an ε > 0 and a sequence of point sets with at least |𝒫|1+ε unit distances. The conjecture is false.
The idea is a generalisation of his own construction, and that is what makes it worth reading. Erdős's grid is the Gaussian integers — points of bounded modulus in the ring of integers of ℚ(i). Everyone who tried to extend it fixed a number field and grew the ball. The machine fixed the ball and grew the field: CM fields of unbounded degree but bounded root discriminant, drawn from an infinite Golod–Shafarevich class field tower. A CM field is the right home because an element of absolute value one in a single embedding has absolute value one in all of them, so unit distances survive the passage to higher degree.
The nine-author note works one explicit case all the way through and lands on 1 + 6.24 × 10−38 — a margin so thin it reads as a joke. But that figure is an illustration, not the result: the note picks T = {3,5,7,11,13,17} and S = {101,∞} “for simplicity” and says of its own class-number bound that it is “not optimal but suffices.”
Optimised, the same construction does far better. Will Sawin’s companion paper — An explicit lower bound for the unit distance problem, arXiv:2605.20579, 20 May 2026 — proves more than n1.014 unit distances. That is a real number, not a rounding error.
Even so, the ceiling stands where Spencer, Szemerédi and Trotter left it in 1984: O(n4/3) ≈ n1.333. Between 1.014 and 1.333 the truth is still unknown. What fell was a belief, not a barrier.
Set the two 2026 results beside each other, because the gallery's whole argument lives in the gap. Problem #728 came back as Lean, and a kernel checked it line by line: the reader was a program, and the verification cost was a compile. Problem #90 came back as prose — correct prose, but prose — and the cost was nine mathematicians spending days digesting five pages into something a referee could hold. Both are real results. Only one of them is cheap to trust.
Melanie Matchett Wood names the hazard on the far side of that gap: it will be "easier for AI to convince humans it has a proof" than to have one. Her counterfactual is sharper still — she thinks the assembled experts would have found this counterexample themselves, had anyone thought to point them at it. What was scarce was never the capability. It was the willingness to spend a month disbelieving Erdős.
One more thing the note records, and it belongs in a gallery about attribution: the machine's write-up did not cite the prior work its own reasoning leaned on — cutting towers of number fields, reflection principles for class group torsion — though it was, in the only sense that applies to a model, familiar with all of it. A human who omitted those citations would be assumed not to have read them. That excuse is no longer available, and the discipline has to become mechanical: attribution is checkable, so check it.
What became of his problems when the collaborator turned machine — the human-checked construction versus the kernel-checked Lean proof, and why the difference is the whole point — is taken up in The Machine Collaborator in Vol V · The Seed.
He died in 1996, at a conference, still working. He had no house, no family of his own, no possessions to speak of — and a share in more theorems than almost anyone who ever lived. The suitcase was empty; the graph was not.
P. Erdős, “On sets of distances of n points,” Amer. Math. Monthly
53 (1946), 248–250.
J. Spencer, E. Szemerédi, W. T. Trotter, “Unit distances in the Euclidean plane,”
Graph Theory and Combinatorics (1984), 293–303.
E. S. Golod, I. R. Shafarevich, “On the class field tower,” Izv. Akad. Nauk SSSR
28 (1964), 261–272.
F. Hajir, C. Maire, R. Ramakrishna, “Cutting towers of number fields,”
Ann. Math. Québec 45 (2021), 321–345.
W. Sawin, “An explicit lower bound for the unit distance problem,”
arXiv:2605.20579 (20 May 2026).
— the optimised exponent, n1.014.
N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang,
M. M. Wood, “Remarks on the disproof of the unit distance conjecture,”
arXiv:2605.20695 (20 May 2026).
T. F. Bloom, Erdős Problem #90,
erdosproblems.com/90.
sorryAx. A clean axiom report is not a reading of the statement: per R20, a theorem can assume its conclusion and still report clean. Follow the link before citing one as evidence.