The Number of Subsets of Integers with Nok-Term Arithmetic Progression

Abstract

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 2 O(r k (n)) , where r k (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 r k (n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(r k (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.

Supersaturated Sparse Graphs and Hypergraphs

“…Keeping this in mind, it seems hopeless to provide a general solution to the counting problem, as it seems crucial to know the order of magnitude of ex(n, H) in order to establish a sufficiently strong supersaturation result. However, Balogh, Liu, and Sharifzadeh [5] have recently managed to settle a question that has a similar flavor without knowing the corresponding extremal function. Specifically, they showed that for infinitely many n, there are 2 Θ(Γ k (n)) many subsets of [n] that do not contain an arithmetic progression of length k; here Γ k (n) is the largest cardinality of a subset of [n] without a k-term arithmetic progression.…”

The Method of Hypergraph Containers

“…This conjecture was disproved by Saxton and Thomason [82], who gave a construction of 2 (1+ε) √ n Sidon sets (for some ε > 0), and also used the hypergraph container method to reprove the following theorem, which was originally obtained in [59] using the graph container method. The second result we would like to state was proved by Balogh, Liu, and Sharifzadeh [10], and inspired the proof presented in Section 4. Let r k (n) be the largest size of a subset of {1, .…”

“…and k ∈ {2, 3, 5}, since in these cases it is known that ex(n, K s,t ) = Θ(n 2−1/s ) and ex(n, C 2k ) = Θ(n 1+1/k ). Very recently, Ferber, McKinley, and Samotij [38], inspired by a similar result of Balogh, Liu, and Sharifzadeh [10] on sets of integers with no k-term arithmetic progression, found a very simple proof of the following much more general theorem. Note that Theorem 4.2 resolves Conjecture 4.1 for every H such that ex(n, H) = Θ(n α ) for some constant α.…”

“…Additive combinatorics. The method of hypergraph containers has been applied to a number of different number-theoretic objects, including sum-free sets [6,7,11,12], Sidon sets [82], sets containing no k-term arithmetic progression [10,13], and general systems of linear equations [82]. (See also [49,50,78] for early applications of the container method to sum-free sets and [28,29,27,59] for applications of graph containers to B h -sets.)…”

The number of multiplicative Sidon sets of integers

“…We claim that both A 2 and A 3 are of small size, |A 2 ∪ A 3 | = n 2/3+o (1) . Thus the contribution of A 2 ∪ A 3 to the number of multiplicative 3-Sidon sets is negligible: n n 2/3+o(1) = 2 o(π(n)) .…”