MathDB

A family of finite sets

\left\{ A_{1},A_{2},.......,A_{m}\right\}

is called equipartitionable if there is a function

\varphi:\cup_{i=1}^{m}

\rightarrow\left\{ -1,1\right\}

such that

\sum_{x\in A_{i}}\varphi\left(x\right)=0

for every

i=1,.....,m.

Let

f\left(n\right)

denote the smallest possible number of

n

-element sets which form a non-equipartitionable family. Prove that a)

f(4k +2) = 3

for each nonnegative integer

k

, b)

f\left(2n\right)\leq1+m d\left(n\right)

, where

m d\left(n\right)

denotes the least positive non-divisor of

n.

Problem Statement