Subquadratic upper bound for isosceles‑triangle‑free subsets of the n×n grid

Prove an upper bound f(n) ≤ n^{1.99} for the function f(n), where f(n) denotes the maximum size of a subset S ⊆ [n]^2 such that no three distinct points a, b, c ∈ S form the vertices of an isosceles triangle (including degenerate/collinear cases, i.e., d(a,b) ≠ d(b,c) for all distinct a,b,c ∈ S with d denoting Euclidean distance).

Background

The problem asks for the maximal number of points one can choose from the n×n grid [n]^2 while avoiding any triple of points forming an isosceles triangle. Writing f(n) for this maximum, a trivial upper bound is f(n) ≤ n^2. Using the observation that horizontal 3-term arithmetic progressions yield isosceles triangles and recent bounds on progression-free sets, this can be improved to f(n) ≤ e^{-c (\log n)^{1/9}} * n^2 for some absolute constant c.

Despite these improvements, any polynomially subquadratic bound remains out of reach. In particular, establishing an upper bound of the form f(n) ≤ n^{1.99} would demonstrate genuinely subquadratic growth, and the authors explicitly note that this remains open.