The lines and planes connecting the points of a finite set
The lines and planes connecting the points of a finite set
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连接有限集的点的直线和平面
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发表时间：
1951
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影响因子：
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通讯作者：
T. Motzkin
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作者：
T. Motzkin
0.1.不迟于1933年，我提出了以下猜想，原来的形式陈述的未成年人的矩阵（1）。T<…。d-空间中任何不在一个超平面上的n个点确定至少n个连通超平面。Ti是微不足道的。很容易看出（4.3）Td是Td_i和Ud的推论d-空间中不在一个超平面上的任何n个点至少确定一个普通超平面，也就是说，一个连通超平面上除了一个点之外的所有给定点都在一个线性（d - 2）-空间上。特别是，T2：n个不共线的点至少由n条直线连接，如果U2成立，则为真：对于n个不共线的点，只有两条直线通过它们。平面三次体的九个子表明，U2在复平面上不成立。然而H. Hanani在1938年给出了一个组合证明T2的每一个（不仅是真实的）投影平面。这个证明的一个大大简化的版本在4.4中给出。1939年A.罗宾逊证明了U2的真实的飞机;在1943年我发现了另一个非常短的证明（分别是第二和第一证明1.1）（2）。在1944年我证明了U3为真实的3-空间（1.4和1.5）。U<…尚未得到证实。在2.2中证明了三条不共点的普通直线的存在，在2.3中证明了四个普通平面的存在。在§3中，确定了对于每个坐标域F，U2成立的n; 3.6包含了对于每个n，U2成立的域F的条件。在§5中对配置进行了应用。最近我得知P. Erdös已经处理了U2（真实的飞机）和T2。U2由西尔维斯特（1893）和埃尔德什（1933）证明，并由T. Gallai（1933）和其他人（我的证明实质上是R. Steinberg（1944））（3）。T2已被G. Szekeres（1940或1941）和N. G. de Bruijn and Erdös（4）.部分的组合
0.1. Not later than 1933 I made the following conjecture, originally in the form of a statement on the minors of a matrix(1). T<¡. Any n points in d-space that are not on one hyper plane determine at least n connecting hyperplanes. Ti is trivial. It is easy to see (4.3) that Td is a consequence of Td_i and UdAny n points in d-space that are not on one hyper plane determine at least one ordinary hyperplane, that is, a connecting hyperplane on which all but one of the given points are on one linear (d — 2)-space. In particular, T2: n not collinear points are connected by at least n straight lines, is true if U2 holds: for n not collinear points there is a straight line through only two of them. Now the nine inflexions of a plane cubic show that U2 does not hold in the complex plane. Nevertheless H. Hanani gave in 1938 a combinatorial proof of T2 for every (not only the real) projective plane. A greatly simplified version of this proof is given in 4.4. In 1939 A. Robinson proved U2 for the real plane; in 1943 I found another very short proof (respectively the second and first proof in 1.1)(2). In 1944 I proved U3 for real 3-space (1.4 and 1.5). U<¡ is still unproved. The existence of three nonconcurrent ordinary lines is proved in 2.2, of four ordinary planes in 2.3. In §3 those n are determined for which U2 holds for every field of coordinates F; 3.6 contains conditions for the fields F for which U2 holds for every n. An application to configurations is made in §5. Recently I learned that P. Erdös had dealt with U2 (for the real plane) and T2. U2 had been conjectured by Sylvester (1893) and by Erdös (1933) and proved by T. Gallai (1933) and others (my proof being substantially that of R. Steinberg (1944)) (3). T2 had been proved by G. Szekeres (1940 or 1941) and by N. G. de Bruijn and Erdös(4). A combination of part