The number of subsets of integers with no k-term arithmetic progression

Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2O(rk(n)), where rk(n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of n, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on rk(n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(rk(n)) contain superlinearly many k-term arithmetic progressions. For integers r and k, Erdős asked whether there is a set of integers S with no (k+1)term arithmetic progression, but such that any r-coloring of S yields a monochromatic k-term arithmetic progression. Nešetřil and Rödl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k ≥ 3 and δ > 0, there exists a reasonably dense subset of primes S with no (k+1)-term arithmetic progression, yet every U ⊆ S of size |U | ≥ δ|S| contains a k-term arithmetic progression. Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea. ∗Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. Email: [email protected]. Research is partially supported by NSA Grant H98230-15-10002, NSF DMS-1500121 and Arnold O. Beckman Research Award (UIUC Campus Research Board 15006). †Mathematics Institute and DIMAP, University of Warwick, Coventry, CV4 7AL, UK. Email: [email protected]. This research was done while HL was at University of Illinois at UrbanaChampaign. ‡Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA. Email: [email protected].