Split the whole numbers from 1 to 2n into two halves of equal size. Slide one half along by some amount and count how many of its members land on members of the other. Some shift always produces a substantial overlap, and Erdős asked in 1955 how small you can force that unavoidable worst case to be.
There is a fact about this problem that makes it much stranger than it looks. Add up the overlap across every possible shift, and the total is always n squared, no matter how you split. Every member of one half meets every member of the other at exactly one shift, so the total is fixed before you start.
So you cannot reduce the overlap. You can only spread it. The entire problem is flattening a distribution whose area is fixed, and the question is how low you can push the peak.
Why obvious splits fail
Put the odds in one half and the evens in the other. Shift by one and they line up exactly: every single member overlaps, the worst possible result. Any block pattern fails the same way, because shifting by one block length maps one half precisely onto the other.
The minimum overlap calculator scores whatever split you give it. The alternating pattern scores 1.0, the worst there is. Blocks of eight score 0.96. A twenty-four character irregular pattern found by random search gets to 0.58. The direction of travel is clear: structure is the enemy, and the good splits look like noise.
The run of improvements
The constant being chased is the limiting value of the worst overlap divided by n. The lower bound is 0.379005, proved by E. P. White in 2022. The upper bound is where the machines have been working.
It had not moved since 2016. In 2025 AlphaEvolve brought it to 0.380924. In 2026 a system called TTT-Discover improved that to 0.380876, and shortly afterwards SimpleTES reached 0.380868. Three different systems, from different groups, each finding a better split than the last.
The gap between the bounds is now about 0.0019, which is to say the answer is known to roughly three decimal places and the argument is over the fourth. It would be easy to be dismissive about that. It would also be wrong: the problem had been stuck for nine years, and the bound is a genuine mathematical object that is now closer to the truth than it was.
What this kind of problem tells you
It is worth noticing what these three systems have in common with the earlier results. The minimum overlap problem has a search space of candidate splits, a scoring function that is cheap to evaluate and completely unambiguous, and an existing record to beat. That is the same shape as the cap set construction in the FunSearch work and the instruction sequences in AlphaDev.
Finding a better object in a scoreable space is the thing machine search does well, and three independent systems converging on the same fourth decimal place is evidence of how well. Proving that no better object exists is an entirely different activity, and the lower bound here is still the one a human proved in 2022.