An uniform version of Dvir and Moran's theorem

Dvir and Moran proved the following upper bound for the size of a family \mbox{\cal F} of subsets of $[n]$ with Vdim(\mbox{\cal F$} \Delta \mbox{$\mathcal{F}$})\leq d$. Let $d\le n$ be integers. Let \mbox{\cal F} be a family of subsets of $[n]$ with Vdim(\mbox{\cal F} \Delta \mbox{\cal F})\leq d. Then \left|$\cal F$\right|\le 2\sum_{k=0}^{\lfloor d/2 \rfloor}\binom nk. Our main result is the following uniform version of Dvir and Moran’s result. Let $d\le n$ be integers. Let \mbox{\cal F} be an uniform family of subsets of $[n]$ with Vdim(\mbox{\cal F} \Delta \mbox{\cal F$})\leq d. Then \[ \left|\mbox{\cal F$}\right|\le 2 {n \choose \lfloor d/2 \rfloor}. ] Denote by ${\mathbf{v}}_{\mathbf{F}}\in \{0,1{\}}^n$ the characteristic vector of a set $F \subseteq [n]$. Our proof is based on the following uniform version of Croot-Lev-Pach Lemma: Let $0\le d\le n$ be integers. Let \mbox{\cal H} be a $k$-uniform family of subsets of $[n]$. Let $\mathbb{F}$ be a field. Suppose that there exists a polynomial $P(x_1, \ldots ,x_n,y_1, \ldots ,y_n)\in \mathbb F[x_1, \ldots ,x_n,y_1, \ldots ,y_n]$with$\mbox{deg}(P)\leq d$suchthat$P(${\mathbf{v}}_{\mathbf{F}}$,${\mathbf{v}}_{\mathbf{F}}$)\ne 0$foreach$F\in \mbox{$\mathcal{H}$}$and$P(${\mathbf{v}}_{\mathbf{F}}$,${\mathbf{v}}_{\mathbf{G}}$)= 0$foreach$F\ne G\in \mbox{$\mathcal{H}$}$. Then \left|$\cal H$\right|\le 2 {n \choose \lfloor d/2 \rfloor}.