Dense integer sets with only logarithmic-length progressions
= Dense integer sets with only logarithmic-length progressions
For any fixed density strictly below one, random subsets of an integer interval can have that density yet avoid <arithmetic progressions> longer than a constant times the logarithm of the interval length. A <union bound> over possible progressions bounds their occurrence, while concentration of the random cardinality supplies a dense realization. Passing to a subset gives an exact prescribed cardinality.