Given a hypergraph $H$, the size-Ramsey number $\hat{ r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{ (3)}_n$ is linear in $n$, i.e., $\hat{ r}_2(P^{ (3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and R\"odl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{ r}_2(P^{ (3)}_n) = O(n^{ 3/2} \log^{ 3/2} n)$.
Categories: math.CO