A note on $k$-wise oddtown problems

Abstract

For integers 2 ≤ t ≤ k, we consider a collection of k set families A j : 1 ≤ j ≤ k whereeven if and only if at least t of the i j are distinct. In this paper, we prove that m = O(n 1/⌊k/2⌋ ) when t = k and m = O(n 1/(t−1) ) when 2t − 2 ≤ k and prove that both of these bounds are best possible. Specializing to the case wherewe recover a variation of the classical oddtown problem.