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We consider the scenario where, for any subset $X' \\subseteq X$ of size $m \\le n$ and for any parameter $1 \\le k \\le m$, the number of restrictions of the sets of $\\S$ to $X'$ of size at most $k$ is only $O(m^{d_1} k^{d-d_1})$, for fixed integers $d > 0$ and $1 \\le d_1 \\le d$ (this generalizes the standard notion of \\emph{bounded primal shatter dimension} when $d_1 = d$). 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The \\emph{discrepancy} of $\\S$ is defined as the minimum of the largest deviation from an even split, over all subsets of $S \\in \\S$ and two-colorings $\\chi$ on $X$. We consider the scenario where, for any subset $X' \\subseteq X$ of size $m \\le n$ and for any parameter $1 \\le k \\le m$, the number of restrictions of the sets of $\\S$ to $X'$ of size at most $k$ is only $O(m^{d_1} k^{d-d_1})$, for fixed integers $d > 0$ and $1 \\le d_1 \\le d$ (this generalizes the standard notion of \\emph{bounded primal shatter dimension} when $d_1 = d$). 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