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We also establish that the $3$-dimensional rectilinear crossing number of a complete $3$-uniform hypergraph having $n \\geq 9$ vertices is at least $\\dfrac{43}{42}\\dbinom{n}{6}$.\n  We prove that the maximum number of crossing pairs of hyperedges in a $4$-dimensional rectilinear drawing of the complete $4$-uniform hypergraph having $n$ vertices is $13\\dbinom{n}{8}$. 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We also establish that the $3$-dimensional rectilinear crossing number of a complete $3$-uniform hypergraph having $n \\geq 9$ vertices is at least $\\dfrac{43}{42}\\dbinom{n}{6}$.\n  We prove that the maximum number of crossing pairs of hyperedges in a $4$-dimensional rectilinear drawing of the complete $4$-uniform hypergraph having $n$ vertices is $13\\dbinom{n}{8}$. 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