  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 $V(G)$. In on-line Ramsey theory, Builder
iteratively presents edges that Painter must immediately color. The $t$-color
on-line size Ramsey number $\tilde R_t (G)$ 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 $\tilde R_t (P_r^{(k)})$. Letting $m=r-k+1$ (the
number of edges in $P_r^{(k)}$), we prove $m^{t-1}/(3\sqrt t)\le\tilde R_t
(P_r^{(2)})\le tm^{t+1}$. For general $k$, a trivial upper bound is ${R \choose
k}$, where $R$ is the least number of vertices in a $k$-uniform (ordered)
hypergraph whose $t$-colorings all contain $P_r^{(k)}$ (and is a tower of
height $k-2$). We prove $R/(k\lg R)\le\tilde R_t(P_r^{(k)})\le R(\lg
R)^{2+\epsilon}$, where $\epsilon$ is any positive constant and $t(m-1)$ is
sufficiently large. Our upper bounds improve prior results when $t$ grows
faster than $m/\log m$. We also generalize our results to $\ell$-loose monotone
paths, where each successive edge begins $\ell$ vertices after the previous
edge.
