Avoiding configurations of small size in the square grid

TL;DR

This paper investigates the maximum size of subsets in an n-by-n grid avoiding specific 3-point and 4-point geometric configurations, providing new bounds and exploring connections with additive combinatorics.

Contribution

It introduces new lower bounds for grid subsets avoiding rhombuses and kites, combining probabilistic methods with additive number theory techniques.

Findings

Rhombus-free subsets have size at least Ω(n^{4/3} (log n)^{-1/3})

Near-quadratic lower bounds for kite-avoiding sets with axis-parallel diagonals

Discussion of implications for square-free sets and geometric configurations

Abstract

We study the maximum size of a subset of the $n \times n$ integer grid that does not contain specific geometric configurations, a variation of the classical problems initiated by Erd\H{o}s and Purdy. While extremal problems for 3-point patterns, such as collinear triples and right triangles are well-studied, the landscape for 4-point configurations in the grid remains less explored. In this paper, we survey the state-of-the-art regarding forbidden 3-point and 4-point configurations, including parallelograms, trapezoids, and concyclic sets. Furthermore, we prove new lower bounds for grid subsets avoiding rhombuses and kites. Specifically, by combining the probabilistic method with the arithmetic properties of Sidon sets, we show that the maximum size of a rhombus-free subset is $\Omega(n^{4/3}(\log n)^{-1/3})$. We also provide near-quadratic lower bounds for sets avoiding kites with axis-parallel diagonals using Behrend-type constructions and discuss implications for square-free sets. These results illustrate the strong interplay between discrete geometry and additive combinatorics.