In Theorem 36 we saw that it is $\mathsf{NP}$-hard to $(1-\delta_0, 1)$-approximate Max-E$3$Sat for some positive but inexplicit constant $\delta_0$. You might wonder how large $\delta_0$ can be. The natural limit here is $\frac18$ because there is a very simple algorithm which satisfies a $\frac78$-fraction of the constraints in any Max-E$3$Sat instance:

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Ryan O'Donnell: Yes, thanks! Sorry for the delay in replying.Ryan O'Donnell: No, it's definitely messed up. Will fix!Ryan O'Donnell: Thanks!Ryan O'Donnell: Hi Amir. It's just a 'dummy variable'; in some sense an X's...Ryan O'Donnell: Right, or I guess I can just switch 'codimension' to 'dimens...Ryan O'Donnell: Yep, thanks!Ryan O'Donnell: Thanks, seems like I spotted this one myself too. (Not corr...