Avoiding configurations of small size in the square grid
Máté Jánosik, Artúr Nádor, Nagy Zoltán Lóránt, László Bence Simon
TL;DR
This work analyzes the maximum size of $[n]^2$ subsets avoiding chosen 3- and 4-point configurations, revealing a rich link between discrete geometry and additive number theory. It combines probabilistic methods with Sidon-type and Behrend-type constructions to derive new lower bounds, notably $\Omega\left(n^{4/3}(\log n)^{-1/3}\right)$ for rhombus-free sets and near-quadratic bounds for kite-free sets with axis-parallel diagonals, while also giving an exact $f_{nd\_para}(n)=2n-1$ result for non-degenerate parallelograms and corollaries for square-free sets. The survey and results cover collinear 4-tuples, parallelograms, isosceles trapezoids, concyclic quadruples, kites, rectangles, and squares, illustrating how geometric symmetry and arithmetic structure of the grid interact to constrain configurations. Overall, the paper advances the understanding of grid-based extremal problems by bridging discrete geometry with additive combinatorics and providing concrete bounds and constructions that push the known landscape forward.
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 Erdős 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 $Ω(n^{4/3}(\log n)^{-1/3})$. 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.