Suppose that is a subset of of maximum cardinality not containing an arithmetic progression of length 3. Let .
How does behave? We do not really know. Will it help talking about it? Can we somehow look beyond the horizon and try to guess what the truth is?
Update 1: for the discussion: Here are a few more specific questions that we can wonder about and discuss.
1) Is the robustness of Behrend’s bound an indication that the truth is in this neighborhood.
2) Why arn’t there analogs for Salem-Spencer and Behrend’s constructions for the cup set problem?
3) What type of growth functions can we expect at all as the answer to such problems?
4) Where is the deadlock in improving the upper bounds for AP-free sets?
5) What new kind of examples one should try in order to improve the lower bounds? Are there some directions that were already extensively explored?
6) Can you offer some analogies? Other problems that might be related?