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Register a new userCharacterization of finite colored spaces with certain conditions
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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.
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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 some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
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-colored graph. We first prove that for all integers $s$ and $t$ with $tgeq sgeq2$, every edge-colored graph $G$ with no properly colored $K_{s,t}$ contains a spanning subgraph $H$ which admits an orientation $D$ such that every directed cycle in $D$ is a properly colored cycle in $G$. Using this result, we show that for $rgeq4$, if the Caccetta-H{a}ggkvist Conjecture holds , then every edge-colored graph of order $n$ with minimum color degree at least $n/r+2sqrt{n}+1$ contains a properly colored cycle of length at most $r$. In addition, we also obtain an asymptotically tight total color degree condition which ensures a properly colored (or rainbow) $K_{s,t}$.
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 isometric sequence of $(X,d)$. In this article we aim to characterize finite metric spaces by their isometric sequences under one of the following assumptions: (i) $a_k=1$ for some $k$ with $2leq kleq n-2$; (ii) $a_k=2$ for some $k$ with $4leq kleq frac{1+sqrt{1+4n}}{2}$; (iii) $a_3=2$; (iv) $a_2=a_3=3$. Furthermore, we give some criterion on how to embed such finite metric spaces to Euclidean spaces. We give some maximum cardinalities of subsets in the $d$-dimensional Euclidean space with small $a_3$, which are analogue problems on a sets with few distinct triangles discussed by Epstein, Lott, Miller and Palsson.
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 $|mathcal{P}| geq (1+varepsilon) q^{lceil d/2 rceil}$, then $mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $mathcal{P} subset mathbb{F}_q^d$ of size $|mathcal{P}| = q^{lceil d/2 rceil}$ that determine at most one spread.
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