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

TL;DR

This paper investigates the number of corner-free subsets in high-dimensional grids, establishing an upper bound related to the maximum size of such sets using advanced combinatorial methods.

Contribution

It introduces a bound on the number of corner-free subsets in grids by applying supersaturation results and the hypergraph container method, advancing understanding of high-dimensional combinatorial structures.

Findings

Number of corner-free subsets is at most exponential in the maximum size of such sets.

Established a supersaturation result for k-dimensional corners.

Applied hypergraph container method to bound the count of corner-free subsets.

Abstract

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.