Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture
Chromatic Graphs, Ramsey Numbers and the Flexible Atom Conjecture
复制标题
色图、拉姆齐数和灵活原子猜想
DOI：
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发表时间：
2008
影响因子：
0.7
通讯作者：
Jacob Manske
中科院分区：
文献类型：
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作者：
Jeremy F. Alm;R. Maddux;Jacob Manske
Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${cal M} subseteq L^{3}$. Let $c : E
ightarrow L$. Let $[n]$ denote the integer set ${1, 2, ldots, n}$. For $x,y,z in V$, let $c(xyz)$ denote the ordered triple $ig(c(xy)$, $c(yz), c(xz)ig)$. We say that $c$ is good with respect to ${cal M}$ if the following conditions obtain: 1. $forall x,y in V$ and $forall (c(xy),j,k) in {cal M}$, $exists z in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $forall x,y,z in V$, $c(xyz) in {cal M}$; and 3. $forall x in V forall ellin L exists , yin V$ such that $ c(xy)=ell $. We investigate particular subsets ${cal M}subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.
Let $K_{N}$ denote the complete graph on $N$ vertices with vertex set $V = V(K_{N})$ and edge set $E = E(K_{N})$. For $x,y in V$, let $xy$ denote the edge between the two vertices $x$ and $y$. Let $L$ be any finite set and ${cal M} subseteq L^{3}$. Let $c : E
ightarrow L$. Let $[n]$ denote the integer set ${1, 2, ldots, n}$. For $x,y,z in V$, let $c(xyz)$ denote the ordered triple $ig(c(xy)$, $c(yz), c(xz)ig)$. We say that $c$ is good with respect to ${cal M}$ if the following conditions obtain: 1. $forall x,y in V$ and $forall (c(xy),j,k) in {cal M}$, $exists z in V$ such that $c(xyz) = (c(xy),j,k)$; 2. $forall x,y,z in V$, $c(xyz) in {cal M}$; and 3. $forall x in V forall ellin L exists , yin V$ such that $ c(xy)=ell $. We investigate particular subsets ${cal M}subseteq L^{3}$ and those edge colorings of $K_{N}$ which are good with respect to these subsets ${cal M}$. We also remark on the connections of these subsets and colorings to projective planes, Ramsey theory, and representations of relation algebras. In particular, we prove a special case of the flexible atom conjecture.