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A colored space is the pair $(X,r)$ of a set $X$ and a function $r$ whose domain is $binom{X}{2}$. Let $(X,r)$ be a finite colored space and $Y,Zsubseteq X$. We shall write $Ysimeq_r Z$ if there exists a bijection $f:Yto Z$ such that $r(U)=r(f(U))$ for each $Uinbinom{Y}{2}$. We denote the numbers of equivalence classes with respect to $simeq_r$ contained in $binom{X}{2}$ and $binom{X}{3}$ by $a_2(r)$ and $a_3(r)$, respectively. In this paper we prove that $a_2(r)leq a_3(r)$ when $5leq |X|$, and show what happens when the equality holds.
Inspired by Andrews 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of
We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces and compute a dynamical invariant attached to their isometry groups.
Properly colored cycles in edge-colored graphs are closely related to directed cycles in oriented graphs. As an analogy of the well-known Caccetta-H{a}ggkvist Conjecture, we study the existence of properly colored cycles of bounded length in an edge-
Let $(X,d)$ be a finite metric space with $|X|=n$. For a positive integer $k$ we define $A_k(X)$ to be the quotient set of all $k$-subsets of $X$ by isometry, and we denote $|A_k(X)|$ by $a_k$. The sequence $(a_1,a_2,ldots,a_{n})$ is called the isome
In this short note, we study the distribution of spreads in a point set $mathcal{P} subseteq mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $varepsilon > 0$, if $|mathcal{P}| geq (1+varepsilon