Isosceles Sets
Isosceles Sets
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等腰组
DOI：
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发表时间：
2009
影响因子：
0.7
通讯作者：
Yury J. Ionin
中科院分区：
文献类型：
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作者：
Yury J. Ionin
1946年，Paul Erdős提出了一个问题，确定一个等腰三角形的最大可能基数，即平面或空间中的一组点，其中任意三个组成一个等腰三角形。这样的问题适用于任何度量空间，而欧几里得空间E的上界（n+ 22）是由Blokhuis[3]找到的。这个上界对于n = 1,2,6和8是很明显的。我们将考虑Erdős关于二进制Hamming空间Hn的问题，并得到Hn的等腰子集S的基数的上界：如果S的点之间最多有两个不同的非零距离，则|S| 6 (n+1 2) +1；更进一步，如果n > 4, n 6= 6，并且作为n立方体的顶点集合，S包含在超平面中，则|S| 6 (n 2)；如果S点之间有两个以上不同的非零距离，则|S |6 (n2) + 1。第一个边界是尖锐的当且仅当n = 2或n = 5；其他两个边界对于所有相关的n值都是锐利的，除了n = 6时的第三个边界，锐利的上限是12。我们还给出了E （n 6 7）的Erdős问题的确切答案，并描述了这些维度中最大基数的所有等腰集。
In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ( n+2 2 ) for the Euclidean space E was found by Blokhuis [3]. This upper bound is known to be sharp for n = 1, 2, 6, and 8. We will consider Erdős’ question for the binary Hamming space Hn and obtain the following upper bounds on the cardinality of an isosceles subset S of Hn: if there are at most two distinct nonzero distances between points of S, then |S| 6 ( n+1 2 ) + 1; if, furthermore, n > 4, n 6= 6, and, as a set of vertices of the n-cube, S is contained in a hyperplane, then |S| 6 ( n 2 ) ; if there are more than two distinct nonzero distances between points of S, then |S| 6 ( n 2 ) + 1. The first bound is sharp if and only if n = 2 or n = 5; the other two bounds are sharp for all relevant values of n, except the third bound for n = 6, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for E with n 6 7 and describe all isosceles sets of the largest cardinality in these dimensions.