L-balancing families

Abstract

P. Hrubeš, S. Natarajan Ramamoorthy, A. Rao and A. Yehudayoff [9] proved the following result: Let p be a prime and let $f\in \mathbb F _p[x_1,\ldots,x_{2p}]$ be a polynomial. Suppose that $f(\mathbf{v_F})=0$ for each $F\subseteq [2p]$, where $|F|=p$ and that $f(\mathbf{0})\neq 0$. Then $\deg (f)\geq p$.

We prove here a generalization of their result for modulo q complete families, where q is a prime power. Let t=2d be an even number and $L\subseteq [d-1]$ be a given subset. We say that $\mathcal{F} \subseteq 2^{[t]}$ is an L-balancing family if for each $F\subseteq [2p]$, where $|F|=d$ there exists a $G\subseteq [n]$ such that $|F\cap G|\in L$.

We give an upper bound for the size of an L-balancing family in the special case if d is a prime.