The Size-Ramsey Number of 3-uniform Tight Paths

Given a hypergraph H, the size-Ramsey number ˆr₂(H) is the smallest integer m such that there exists a hypergraph 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 Pn⁽³⁾ is linear in n, i.e., ˆr₂(Pn⁽³⁾) = O(n). This answers a question by Dudek, La Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417–434], who proved ˆr₂(Pn⁽³⁾) = O(n^3/2 log^3/2 n).