Erdős Problem #192

Let \(A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d\) be an infinite sequence such that \(a_{i+1}-a_i\) is a positive unit vector (i.e. is of the form \((0,0,\ldots,1,0,\ldots,0)\)). For which \(d\) must \(A\) contain a three-term arithmetic progression? This is true for \(d\leq 3\) and false for \(d\geq 4\). This problem is equivalent to one on 'abelian squares' (see [231](https://www.erdosproblems.com/231)). In particular \(A\) can be interpreted as an infinite string over an alphabet with \(d\) letters (each letter describining which of the \(d\) possible steps is taken at each point). An abelian square in a string \(s\) is a pair of consecutive blocks \(x\) and \(y\) appearing in \(s\) such that \(y\) is a permutation of \(x\). The connection comes from the observation that \(p,q,r\in A\subset \mathbb{R}^d\) form a three-term arithmetic progression if and only if the string corresponding to the steps from \(p\) to \(q\) is a permutation of the string corresponding to the steps from \(q\) to \(r\). This problem is therefore equivalent to asking for which \(d\) there exists an infinite string over \(\{1,\ldots,d\}\) with no abelian squares. An infinite string without abelian squares was constructed when \(d=4\) by Keränen [Ke92]. W