Treffer: On Combinatorial Cubes: On combinatorial cubes

Given \(k\geq 1\), a set \(H\subset \mathbb{N}\) is called a cube of size \(k\) if there exist \(a>0\) and \(x_1,\dots,x_k\) such that \(H=\left\{a+\sum_{i=1}^k\varepsilon_ia_i:\varepsilon=0\text{ or }1\right\}\) and we write \(\dim H=k\). Let \(r_3(n)\) be the maximal number of integers that can be selected from the interval \([1,n]\) without including a three--term arithmetic progression. The author proves that there exists a \(A\subset[1,n]\) for which \(| A| \geq r_3(n)/3\) and \(\max_{H\subset[1,n]}\dim H\leq (1/\log 2)\log\log 2\).