The number of $k$-dimensional corner-free subsets of grids

A subset $A$ of the $k$-dimensional grid ${1,2, cdots, N}^k$ is called $k$-dimensional corner-free if it does not contain a set of points of the form ${ a } cup { a + de_i : 1 leq i leq k }$ for some $a in {1,2, cdots, N}^k$ and $d > 0$, where $e_1,e_2, cdots, e_k$ is the standard basis of $mathbb{R}^k$. We define the maximum size of a $k$-dimensional corner-free subset of ${1,2, cdots, N}^k$ by $c_k(N)$. In this paper, we show that the number of $k$-dimensional corner-free subsets of the $k$-dimensional grid ${1,2, cdots, N}^k$ is at most $2^{O(c_k(N))}$ for infinitely many values of $N$. Our main tool for the proof is a supersaturation result for $k$-dimensional corners in sets of size $Theta(c_k(N))$ and the hypergraph container method.