Avoiding configurations of small size in the square grid

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.

💡 Research Summary

The paper investigates extremal subsets of the n × n integer grid that avoid prescribed small geometric configurations, extending the classical Erdős–Purdy framework. After a brief historical overview of 3‑point problems—no‑3‑in‑line, isosceles and right‑isosceles triangles—the authors turn to 4‑point patterns. They survey known results for collinear 4‑tuples (f₄‑coll ≤ 3n, f₄‑coll ≥ 1.973 n), parallelograms (the two‑dimensional Sidon bound f_para = n + O(n^{2/3}) and an exact value 2n − 1 when only non‑degenerate parallelograms are forbidden), and concyclic quadruples (upper bound 2.5 n − 1, lower bound ≈0.58 n).

The main contributions are new lower bounds for rhombus‑free and kite‑free sets. For rhombus‑free subsets, the authors select a Sidon set S⊂