Integer points avoiding three on a line, four on a circle

Question

A century ago, Dudeney asked to place $16$ pawns on a chessboard with no three on a line:

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As described by David Eppstein,¹ the maximum number $g_3(n)$ points that can be placed on an $n \times n$ integer grid, avoiding three points on a line, satisfies $$ \tfrac{3}{2} n - o(n) \le g_3(n) \le 2 n \;.$$

My question is:

Q. What are bounds on $g_4(n)$, the maximum number of points that can be placed on an $n \times n$ integer grid, no three on a line, no four cocircular?

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          ^{Perhaps no $4$ of the $6$ points (right figure) are cocircular?
          Added: @AaronMeyerowitz shows that in fact $4$ points are cocircular.}

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The question can be generalized to avoiding points lying on an algebraic curve of degree $d$.

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¹ Eppstein, David. Forbidden Configurations in Discrete Geometry. Cambridge University Press, 2018. Chapter 9, pages 72–75. Cambridge link.

Answer 1

this should be considered as a comment:

There seems to be a strong relation of the "no 4 points cocircular" problem to pythagorean triples:

Taking $(0,0)$ as one of the corners of Pythagorean triangles excluding that corner from the pointset to be generated, yields eight cocircular gridpoints, of which not more that three can be in the final pointset, thus yielding as a first upper bound marked points, namely $\frac{3}{2}\#PT_L$ where $PT_L$ denotes the number of Pythagorean triples $\lbrace (a,b,c)\ |\ 0\lt a\lt b\lt c\le L\rbrace$ with given upper bound $L$ on the length of the hypotenuse $c$.

A further refinement of the estimate is possible by taking into account the number of different (primitive) Pythagorean triples with equal length of the hypotenuse; that number seems to be $2^{k-1}$, where $k$ is the number of $c$'s primefactors of the form $4p+1$ (cf e.g. number of primitive pythagorean triples with common hypotenuse.

So, because of the apparent relation to number theoretic questions, there seems little hope to find sharp upper bounds on the maximal number of points in a square grid, of which no four are co-circular.