Progression-free sets in cyclic groups

How large can a subset of ℤ/nℤ be if it contains no three distinct terms in arithmetic progression?

Established so far

The eight ternary-digit residues modulo 27 contain no three distinct terms in progression.

The same digit rule fails modulo 4.

Ideas

Use ternary digits to build a large set modulo 3ᵏSample ideaThe eight numbers with ternary digits 0 or 1 are progression-free modulo 27 when terms must be distinct.Open idea ›

Extend the digit construction to every modulusSample ideaThe direct extension already fails modulo 4: (3, 0, 1) is a nontrivial progression.Open idea ›

Compare the construction with the density boundSample ideaThe finite examples do not settle the asymptotic density. A precise upper bound is still needed.Open idea ›

Randomly thin a progression-free setSample ideaA small search found no gain over the digit construction through modulus 27.Open idea ›

Count differences inside a translated setSample ideaThe first count is exact for modulus 19, but the general estimate is not sharp.Open idea ›

Classify the smallest exceptional moduliSample ideaThe checked cases split by parity; no uniform explanation has emerged.Open idea ›

Look for a construction beyond ternary digitsSample ideaA search over two small digit patterns found no larger set modulo 27.Open idea ›