2026/07/23 by Xiaolei Niu, Yinghui Hang, Haitao Cao
#math.CO
In this paper, we study the supersaturation problems of eventown. Given a family A of subsets of an n element set, let op(A) denote the number of distinct pairs A,B∈ A for which |A∩ B| is odd. We give extremal eventown constructions and show that for fixed s≤2\lfloor (n)/(2) \rfloor-2, there exists a collection of 2\lfloor(n)/(2)\rfloor+s even-sized subsets of an n element set that contains exactly s⋅ 2\lfloor (n)/(2) \rfloor-1 pairwise intersections of odd size. This extends the range of s in a conjecture proposed by O'Neill from 2\lfloor (n)/(2) \rfloor-2\lfloor (n)/(4) \rfloor to 2\lfloor (n)/(2) \rfloor-2. We also give a construction using symmetric designs to prove that when k is even and 4k-1 is a prime power, there exists a collection of 2\lfloor(4k-1)/(2)\rfloor+s even-sized subsets of a 4k-1 element set As with op(As)=s ⋅ 2^\lfloor(4k-1)/(2)\rfloor-1, 1≤ s≤4k-1.