O’Neill’s supersaturation conjecture for eventown families

Establish that every family of even-sized subsets of [n] with cardinality 2^{\lfloor n/2\rfloor}+s, where 1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}, contains at least s\,2^{\lfloor n/2\rfloor-1} unordered pairs whose intersection has odd cardinality.

Background

An eventown family is a collection of even-sized subsets of [n] such that every pair of distinct members has an even-sized intersection. The eventown theorem gives the extremal bound 2^{\lfloor n/2\rfloor} for the size of such a family. The supersaturation problem asks how many odd-intersection pairs must occur when a family exceeds this bound by an excess of s members.

The conjecture predicts the optimal lower bound s\,2^{\lfloor n/2\rfloor-1} throughout the range 1\le s\le 2^{\lfloor n/2\rfloor}-2^{\lfloor n/4\rfloor}. The paper proves the bound only for s up to a fixed positive proportion, namely s\le 2^{\lfloor n/2\rfloor}/26, so the conjecture remains unresolved over the remainder of its stated range.