Gap-ETH

A $3$-SAT formula is called satisfiable if all its clauses can be satisfied, and it has at most a $1-\epsilon$ fraction of satisfiable clauses if no assignment satisfies more than that fraction. Here $n$ denotes the number of clauses.

Gap-ETH. There exist constants $\epsilon,c>0$ such that no algorithm running in time $O(2^{cn})$ can distinguish between a satisfiable $3$-SAT formula and a $3$-SAT formula with at most a $1-\epsilon$ fraction of satisfiable clauses.

Gap-ETH is a stronger quantitative variant of the Exponential Time Hypothesis used to establish algorithmic lower bounds. Under ordinary ETH, Dinur's quasilinear-length PCP gives a weaker lower bound; a linear-length PCP would imply Gap-ETH from ETH.