For an index \(i : [d]\), we define

We say a tuple \((a_1, \dots , a_d) : (G^d)^d\) is a forbidden pattern with tip \(v : G^d\) if

for all \(i, j\) distinct. We also simply say \((a_1, \dots , a_d)\) is a forbidden pattern if it is a forbidden pattern with tip \(v\) for some \(v\).

A multidimensional corner in \(d\) dimensions is a tuple of the form \((x, x + ce_1, \dots , x + ce_d)\) for some \(x : G^d\) and \(c : G\), where \(ce_i\) is the vector of all zeroes except in position \(i\) where it is \(c\). Such a corner is said to be trivial if \(c = 0\).

The \(d\)-dimensional corner-free number of \(G\), denoted \(r_d(G)\) is the size of the largest set \(A\) in \(G^d\) such that \(A\) doesn’t contain a non-trivial corner.

The \(d\)-dimensional corner-coloring number of \(G\), denoted \(\chi _d(G)\), is the smallest number of colors one needs to color \(G^d\) such that no non-trivial \(d\)-dimensional corner is monochromatic.