For any fixed \(d\ge 1\) and subset X of \(\mathbb {N}^d\) , let \(r_X(n)\) be the maximum cardinality of a subset A of \(\{1,\dots,n\}^d\) which does not contain a subset of the form \(\varvec{b} + rX\) for \(r>0\) and \(\varvec{b} \in \mathbb {R}^d\) . Such a set A is said to be X-free. The Multidimensional Szemerédi Theorem of Furstenberg and Katznelson states that \(r_X(n)=o(n^d)\) . We show that, for \(|X|\ge 3\) and infinitely many \(n\in \mathbb {N}\) , the number of X-free subsets of \(\{1,\dots,n\}^d\) is at most \(2^{O(r_X(n))}\) . The proof involves using a known multidimensional extension of Behrend’s construction to obtain a supersaturation theorem for copies of X in dense subsets of \([n]^d\) for infinitely many values of n and then applying the powerful hypergraph container lemma. Our result generalizes work of Balogh, Liu, and Sharifzadeh on k-AP-free sets and Kim on corner-free sets.
