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On distinct distances in homogeneous sets in the Euclidean space
A homogeneous set of points in the -dimensional Euclidean space
determines at least distinct distances
for a constant . In three-space, we slightly improve our general bound
and show that a homogeneous set of points determines at least
distinct distances
Bounds on three- and higher-distance sets
A finite set X in a metric space M is called an s-distance set if the set of
distances between any two distinct points of X has size s. The main problem for
s-distance sets is to determine the maximum cardinality of s-distance sets for
fixed s and M. In this paper, we improve the known upper bound for s-distance
sets in n-sphere for s=3,4. In particular, we determine the maximum
cardinalities of three-distance sets for n=7 and 21. We also give the maximum
cardinalities of s-distance sets in the Hamming space and the Johnson space for
several s and dimensions.Comment: 12 page
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
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