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A \textit{colored space} is {a pair} $(X,r)$ of a set $X$ and a function $r$ whose domain is $X\choose 2$. Let $(X,r)$ be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that $r(U)=r(f(U))$ for each $U\in {Y\choose 2}$ {where $f(U)=\{f(u)\mid u\in U\}$}. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in {$X \choose i$ by $a_i(r)$}. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens {when equality} holds.
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참고문헌 (5)
참고문헌 신청
E. Bannai / 1983 / An upper bound for the cardinality of an s-distance subset in real Euclidean space. II / Combinatorica 3(2):147~152
S. J. Einhorn / 1966 / On euclidean sets having only two distances between points. II / Nederl. Akad. Wetensch. Proc. Ser. A 69(28):489~504
M. Hirasaka / Characterization of finite metric spaces by their isometric sequences / arXiv:1802.06097
D. G. Larman / 1977 / On two-distance sets in Euclidean space / Bull. Lond. Math. Soc 9(3):261~267
P. Lisonek / 1997 / New maximal two-distance sets / J. Combin. Theory Ser. A 77(2):318~338
S. J. Einhorn / 1966 / On euclidean sets having only two distances between points. II / Nederl. Akad. Wetensch. Proc. Ser. A 69(28):489~504
M. Hirasaka / Characterization of finite metric spaces by their isometric sequences / arXiv:1802.06097
D. G. Larman / 1977 / On two-distance sets in Euclidean space / Bull. Lond. Math. Soc 9(3):261~267
P. Lisonek / 1997 / New maximal two-distance sets / J. Combin. Theory Ser. A 77(2):318~338
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