ETH-Hardness of $Gap\text-MAX\text-3SAT_{c}$ The PCP theorem can be stated like this : 

There is a polynomial time reduction from SAT to  $Gap\text-MAX\text-3SAT_{c}$ i.e. there is a reduction that maps an instance $\phi$ of SAT to an instance $\psi$ of  $Gap\text-MAX\text-3SAT_{c}$ such that 


If $\phi$ is satisfiable, so is $\psi$.
If $\phi$ is unsatisfiable, then $OPT(\psi) \leq c$, i.e., there is no assignment to the variables of $\psi$ that satisfies more than a fraction $c$ of the  clauses.


What is the blow-up of this reduction? That is, what are the parameters of the instanceof  $Gap\text-MAX\text-3SAT_{c}$ in terms of parameter of the given instance of SAT. Is the blow-up linear?
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