  For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat
R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest
integer $m$ such that there exists a $k$-uniform hypergraph $\mathcal{H}$ on
$m$ edges such that any coloring of the edges of $\mathcal{H}$ with $r$ colors
yields a monochromatic copy of $\mathcal{G}$. Let $\mathcal{P}_{n,k-1}^{(k)}$
denote the $k$-uniform tight path on $n$ vertices. Dudek, Fleur, Mubayi and
R\H{o}dl showed that the size-Ramsey number of tight paths $\hat
R(\mathcal{P}_{n,k-1}^{(k)}, 2) = O(n^{k-1-\alpha} (\log n)^{1+\alpha})$ where
$\alpha = \frac{k-2}{\binom{k-1}{2}+1}$. In this paper, we improve their bound
by showing that $\hat R(\mathcal{P}_{n,k-1}^{(k)}, r) = O(r^k (n\log n)^{k/2})$
for all $k\geq 3$ and $r\geq 2$.
