Let \({\mathscr {S}}\) be a set of n points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of \({\mathscr {S}}\) is less than \(Kn^3\) for some \(K=o(n^{{1}/{7}})\) then, for n sufficiently large, all but at most O(K) points of \({\mathscr {S}}\) are contained in the intersection of five linearly independent quadrics. Conversely, we prove that there are finite subgroups of size n of an elliptic curve that span less than \(n^3/6\) solids containing exactly four points of \({\mathscr {S}}\).

令\（{\ mathscr {S}} \）为实四维空间中的n个点的集合，没有四个共面且跨越整个空间。我们证明，如果固体入射与恰好四个点的数量\（{\ mathscr {S}} \）小于\（KN ^ 3 \）对于一些\（K = O（N ^ {{1} / {7}}）\），然后，对于n足够大的情况，\（{\ mathscr {S}} \）的所有O（K）点（至多O（K）点）都包含在五个线性独立二次曲面的交点中。相反，我们证明了椭圆曲线的大小为n的有限子组，其跨度小于\（n ^ 3/6 \）恰好包含\（{\ mathscr {S}} \）四点的实体。