A Note on k-Wise Oddtown Problems

For integers $$2 \le t k$$ , we consider a collection of k set families $${\mathcal {A}}_j: 1 j where {A}}_j = \{ A_{j,i} \subseteq [n] : i m \}$$ and $$|A_{1, i_1} \cap \cdots A_{k,i_k}|$$ is even if only at least the $$i_j$$ are distinct. In this paper, prove that $$m =O(n^{ 1/ \lfloor k/2 \rfloor })$$ when $$t=k$$ $$m=O(n^{1/(t-1)})$$ $$2t-2 both these bounds best possible. Specializing to case {A}}= {\mathcal {A}}_1 {A}}_k$$ recover variation classical oddtown problem.