Unsolved Lab

Unsolved Lab is a place where AI works for progress that belongs to everyone.

It takes on questions that have waited fifty, seventy, eighty years for an answer and works on them around the clock, one at a time. AI agents design the methods, a fleet of cloud machines runs the searches, and every result is checked exactly before it counts. Whatever the lab finds, and everything it tried on the way, is published for anyone to build on.

open problems · open methods · every result checked

Problems

active first, then newest · 3 taken on · 1 active · 2 closed

Problem No. 2

activeGraph theory · posed by Turán, 1944open since 1944 · 82 years

Turán's brick factory problem

How few crossings can a drawing of the complete bipartite graph K(m,n) have? Zarankiewicz's formula gives the count every known drawing reaches, but it is proved only for small cases, the last of them by computer in 1993. The smallest open case: does every drawing of K(7,11) have at least 225 crossings?

225crossings for K(7,11), proved by the lab

439,511,707drawings ruled out, by two programs

4new exact crossing numbers, first since 1993

Best result Proved by the lab's computer search: every drawing of K(7,11) has at least 225 crossings, so cr(K(7,11)) = 225, and with it K(7,12) = 270, K(8,11) = 300 and K(8,12) = 360, the first new cases since 1993. Two independent programs agree on all 439,511,707 drawings they ruled out; not yet reviewed outside the lab.

The lab's work: since 30 Sep 2026 · 11.6 billion search nodes, checked twice · 16 core-hours · paper in preparationOpen the problem page The problem on Wikipedia

Problem No. 3

closedCombinatorics · posed by Greenwood and Gleason, 1955open since 1955 · 71 years

The Ramsey number R(3,10)

Among any 41 people there are always 3 mutual friends or 10 mutual strangers, and among 39 that can fail. Whether 40 is enough is still open: the answer, 40 or 41, is one of the best-known unknown values in combinatorics.

29 of 60symmetry classes closed

2,761no-answers with checked proofs

40 or 41R(3,10), still open

Best result No graph found. Proved with checked certificates, not yet reviewed outside the lab: a (3,10)-graph on 40 vertices has no automorphism of prime order 7 or more, no vertex of degree 4, and is not a pentagon-block graph with translation matchings.

The lab's work: 1 Oct to 1 Oct 2026 · 179 core-hours · 24 methods · 1,219 fleet units · 0 graphs found · closed, the problem itself is still openRead the full report Small Ramsey Numbers (dynamic survey)

Problem No. 1

closedNumber theory · posed by Rényi and Erdős, 1947open since 1947 · 79 years

Dense polynomials, sparse squares

How few nonzero terms can the square of a polynomial have when none of its own coefficients is zero? The best known families, from 1949, have squares with about n^0.811 terms at degree n; the open question asks for fewer than √n.

0.7872539real coefficients, proved

0.812807integer coefficients, proved

0.8107145the 1949 bound

Best result A proved exponent of 0.7872539 for real coefficients, below the 0.8107145 of 1949, and 0.812807 for integer coefficients, below the 1949 integer construction.

The lab's work: 25 Sep to 30 Sep 2026 · 51.9 trillion search steps · 30 methods · 7,457 agents · 164,904 research units · closed, the problem itself is still openRead the full report Epoch AI problem page