> a limit on the size of a set of integers in which no three of them are evenly spaced

This misses a key detail. You can trivially find arbitrarily large such sets e.g. take the first however many powers of 2: 1, 2, 4, 8, 16 ...

The missing constraint is that the set of integers must be a subset of { 1, 2, ... , N }.

> Erdős and Turán wanted to know how many numbers smaller than some ceiling N can be put into a set without creating any three-term arithmetic progressions.