O’Neill’s eventown supersaturation conjecture

O’Neill conjectured that the standard construction minimizes odd-intersection pairs among sufficiently large families of even-sized subsets. The conjecture is known in only part of its proposed parameter range.

Known results

• O’Neill (2022): the sharp bound holds for $s\in\{1,2\}$.
• Wei, Zhao, Zhang, and Ge (2023): for sufficiently large $n$, it holds when $1\le s\le 2^{\lfloor n/8\rfloor}/n$.
• A general Fourier-analytic argument gives the weaker bound $op(\mathcal A)\ge s\,2^{\lfloor n/2\rfloor-2}$.
• Niu, Hang, and Cao (2026) extended the matching construction, but not the corresponding lower-bound proof, to $s\le 2^{\lfloor n/2\rfloor}-2$.

August 2026 fixed-proportion extension

A new arXiv item reports that the sharp lower bound is proved for a fixed positive proportion of the extremal eventown size, substantially enlarging the known range. The supplied record does not provide authors or enough theorem detail for independent verification.

Current status (as of August 2026): the sharp bound is established for $s\in\{1,2\}$, for $s\le 2^{\lfloor n/8\rfloor}/n$ when $n$ is sufficiently large, and reportedly for a newer fixed-proportion range, while the full conjectured range remains open.