Abstract: An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices. The appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number $R'_t(G)$ 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 $P(r,k)$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices. We obtain good bounds on $R'_t(P(r,k))$. Letting $m=r-k+1$ (the number of edges in $P(r,k)$), we prove $m^{t-1}/(3\sqrt t) \le R'(P(r,2)) \le tm^{t+1}$. For general $k$, a trivial upper bound is $\left({N \atop k}\right)$, where $N$ is the vertex Ramsey number of $P(r,k)$ and is a tower of height $k-2$. We prove $N/(k\log N) \le R'_t(P(r,k)) \le N(\log N)^{2+c}$, where $c$ is any positive constant and $t(m-1)$ is sufficiently large. Our upper bounds improve prior results when $t$ grows faster than $m/\log m$, and our methods yield another derivation of the vertex Ramsey number. We also generalize our results to $l$-loose monotone paths, where each successive edge begins $l$ vertices after the previous edge (the tight path is 1-loose). This work is joint with Xavier Pérez Giménez and Pawel Pralat.