A note on rainbow saturation number of paths

For a fixed graph $F$ and an integer $t$, the dfn{rainbow saturation number} of $F$, denoted by $sat_t(n,mathfrak{R}(F))$, is defined as the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a dfn{rainbow copy} of $F$, i.e., a copy of $F$ all of whose edges receive a different color, but the addition of any missing edge in any color from $[t]$ creates such a rainbow copy. Barrus, Ferrara, Vardenbussche and Wenger prove that $sat_t(n,mathfrak{R}(P_ell))ge n-1$ for $ellge 4$ and $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell-1} rceil cdot binom{ell-1}{2}$ for $tge binom{ell-1}{2}$, where $P_ell$ is a path with $ell$ edges. In this short note, we improve the upper bounds and show that $sat_t(n,mathfrak{R}(P_ell))le lceil frac{n}{ell} rceil cdot left({{ell-2}choose {2}}+4right)$ for $ellge 5$ and $tge 2ell-5$.