An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V\left(G\right)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number ${\tilde{R}}_t\left(G\right)$ of an ordered hypergraph $G$ is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using $t$ colors to produce a monochromatic copy of $G$. The monotone tight path $P_r^{\left(k\right)}$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices.

We obtain good bounds on ${\tilde{R}}_t\left(P_r^{\left(k\right)}\right)$. Letting $m=r-k+1$ (the number of edges in $P_r^{\left(k\right)}$), we prove $m^{t-1}∕\left(3\sqrt{t}\right)\le {\tilde{R}}_t\left(P_r^{\left(2\right)}\right)\le t{m}^{t+1}$. For general $k$, a trivial upper bound is $\left(\genfrac{}{}{0ex}{}{R}{k}\right)$, where $R$ is the least number of vertices in a $k$-uniform (ordered) hypergraph whose $t$-colorings all contain $P_r^{\left(k\right)}$ (and is a tower of height $k-2$). We prove $R∕(klgR)\le {\tilde{R}}_t\left(P_r^{\left(k\right)}\right)\le R{(lgR)}^{2+\epsilon }$, where $\epsilon$ is a positive constant and $tm$ is sufficiently large in terms of ${\epsilon }^{-1}$. Our upper bounds improve prior results when $t$ grows faster than $m∕logm$. We also generalize our results to $\ell$-loose monotone paths, where each successive edge begins $\ell$ vertices after the previous edge.

一个有序的超图是一个超图$H$ 具有指定的顶点线性顺序和有序超图的外观 $G$ 在 $H$ 必须遵守指定的命令 $V（G）$。在在线Ramsey理论中，Builder迭代地呈现Painter必须立即着色的边缘。的$Ť$色在线尺寸拉姆齐数 ${\widetilde{\mathrm{[R}}}_{Ť}（G）$ 有序超图 $G$ 是Builder强制使用Painter使用（在大量有序的一组顶点上）播放的最小边数 $Ť$ 颜色以产生的单色副本 $G$。该单调紧张的路径 $P_{\mathrm{[R}}^{（ķ）}$ 是有序超图 $\mathrm{[R}$ 顶点均为边集的顶点 $ķ$ 连续的顶点。

我们在 ${\widetilde{\mathrm{[R}}}_{Ť}（P_{\mathrm{[R}}^{（ķ）}）$。出租$米=\mathrm{[R}-ķ+\mathrm{1个}$ （中的边数 $P_{\mathrm{[R}}^{（ķ）}$），我们证明 ${米}^{Ť-\mathrm{1个}}∕（3\sqrt{Ť}）\le {\widetilde{\mathrm{[R}}}_{Ť}（P_{\mathrm{[R}}^{（2）}）\le Ť{米}^{Ť+\mathrm{1个}}$。对于一般$ķ$，一个很小的上限是 $\left(\genfrac{}{}{0ex}{}{\mathrm{[R}}{ķ}\right)$，在哪里 $\mathrm{[R}$ 是一个顶点中最少数量的顶点 $ķ$-一致（有序）超图，其 $Ť$-所有颜色都包含 $P_{\mathrm{[R}}^{（ķ）}$ （并且是高塔 $ķ-2$）。我们证明$\mathrm{[R}∕（ķlg\mathrm{[R}）\le {\widetilde{\mathrm{[R}}}_{Ť}（P_{\mathrm{[R}}^{（ķ）}）\le \mathrm{[R}{（lg\mathrm{[R}）}^{2+\epsilon }$，在哪里 $\epsilon$ 是一个正常数， $Ť米$ 就...而言足够大 ${\epsilon }^{-\mathrm{1个}}$。在以下情况下，我们的上限可以改善先前的结果$Ť$ 增长快于 $米∕日志米$。我们还将结果推广到$\ell$-松散的单调路径，其中每个连续边开始 $\ell$ 上一条边之后的顶点。