ETH-Hardness of $Gap\text-MAX\text-3SAT_{c}$
I think the best known result is that the blow-up can be quasi-linear (the new instance has size $n\cdot(\log n)^{O(1)}$). This is given in Dinur's 2007 paper (Thm 8.1), which is also cited by the Moshkovitz-Raz paper mentioned by Yuval Filmus. 

As far as I know it is still open if the blow-up can be brought down to linear. Some people call this the Linear PCP conjecture, see e.g. the recent paper by Bonnet, Egri and Marx