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on Subsets of the grid with no isosceles triangles (AlphaEvolve problem 59)

The statement

For $n$ a natural number, let $C(n)$ denote the size of the largest subset of $[n]^2 = \{1,\dots,n\}^2$ that does not contain a (possibly flat) isosceles triangle. In other words,

C(n) \coloneqq \max_{S\subset [n]^2}\{|S|: a,b,c\in S \text{ distinct} \implies \|a-b\| \neq \|b-c\|\}.

Obtain upper and lower bounds for $C(n)$ that are as strong as possible.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.