Title: Planar point sets with forbidden $4$-point patterns and few distinct distances Show Chinese title

Title: 平面点集禁止$4$点模式和少量不同距离

Abstract: We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a question of Erd\H{o}s. The proof combines an analysis by Dumitrescu of forbidden four-point patterns with an algebraic construction of Thiele and Dumitrescu (to eliminate parallelograms), as well as a randomized transformation of that construction (to eliminate most other forbidden patterns). Show Chinese abstract

Abstract: 我们证明，对于任何大的$n$，存在平面上的一组$n$个点，有$O(n^2/\sqrt{\log n})$个不同的距离，使得该集合中的任意四个点至少确定五个不同的距离。 这否定了Erdős提出的一个问题。 证明结合了Dumitrescu对禁止的四点模式的分析，以及Thiele和Dumitrescu的代数构造（以消除平行四边形），以及对该构造的随机变换（以消除大多数其他禁止的模式）。