论文标题
在有序的拉姆西数量的三方3-均匀的超图
On ordered Ramsey numbers of tripartite 3-uniform hypergraphs
论文作者
论文摘要
对于整数$ k \ geq 2 $，订购的$ k $均匀的hypergraph $ \ mathcal {h} =（h，h，<）$是$ k $ - 均匀的hypergraph $ h $，以及固定的线性订购$ <$ <$ <$ <$ <$ <$ <$。订购的Ramsey数字$ \叠加{r}（\ Mathcal {h}，\ Mathcal {g}）$的两个有序$ k $ - 均匀的超差异$ \ nathcal {h} $ {h} $ and $ \ nathcal {g} $ in \ n. n plode of \ mathbb { $ k $ - 统一超graph $ \ Mathcal {k}^{（k）} _ n $ on $ n $ vertices包含$ \ Mathcal {h} $的蓝色副本或$ \ Mathcal {g} $的红色副本。 有序的Ramsey数量进行了有序的图表，但对于较高均匀性的有序超图表知之甚少。我们对有序的3-均匀超图的有序数量的Ramsey数量提供了一些第一个非平地估计。特别是，我们证明，对于所有$ d，n \ in \ mathbb {n} $，对于每一个订购的$ 3 $ - 均匀的超graph $ \ mathcal { $$ \ overline {r}（\ Mathcal {h}，\ Mathcal {h}）\ leq 2^{o（n^{2- \ \ varepsilon}）}。 $ \ OVERLINE {r}（\ MATHCAL {g}，\ MATHCAL {K}^{（3）} _ 3（n））$，其中$ \ Mathcal {G} $是订购的3-均匀刻度，带有$ n $ n $ n $ Vertices和最高$ d $ d $ d $ d $ d $ d $ d $ d $ d undere的$ d undert and，三方超图和连续的颜色类别的尺寸$ n $。我们证明，通过证明$ \ overline {r}（\ Mathcal {h}，\ Mathcal {k}^{（3）} _ 3（n））\ geq 2^{ω（n \ log {n}} $ 3 $ 3 $ 3 $ 3 $ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ caph
For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$ of two ordered $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$ is the smallest $N \in \mathbb{N}$ such that every red-blue coloring of the hyperedges of the ordered complete $k$-uniform hypergraph $\mathcal{K}^{(k)}_N$ on $N$ vertices contains a blue copy of $\mathcal{H}$ or a red copy of $\mathcal{G}$. The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all $d,n \in \mathbb{N}$ and for every ordered $3$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $3$ there is an $\varepsilon=\varepsilon(d)>0$ such that $$\overline{R}(\mathcal{H},\mathcal{H}) \leq 2^{O(n^{2-\varepsilon})}.$$ In fact, we prove this upper bound for the number $\overline{R}(\mathcal{G},\mathcal{K}^{(3)}_3(n))$, where $\mathcal{G}$ is an ordered 3-uniform hypergraph with $n$ vertices and maximum degree $d$ and $\mathcal{K}^{(3)}_3(n)$ is the ordered complete tripartite hypergraph with consecutive color classes of size $n$. We show that this bound is not far from the truth by proving $\overline{R}(\mathcal{H},\mathcal{K}^{(3)}_3(n)) \geq 2^{Ω(n\log{n})}$ for some fixed ordered $3$-uniform hypergraph $\mathcal{H}$.
