Largest Sidon subset guaranteed in every N-point real set: is $\ell(N)\sim N^{1/2}$? (Erdős #530)

For a finite set $A\subset\mathbb{R}$, call $S\subseteq A$ a Sidon set if the only solutions to $a+b=c+d$ with $a,b,c,d\in S$ are the trivial ones (those with $\{a,b\}=\{c,d\}$). Let $\ell(N)$ be the largest integer such that every $A\subset\mathbb{R}$ with $\lvert A\rvert=N$ contains a Sidon subset $S$ with $\lvert S\rvert\geq\ell(N)$. Determine the order of growth of $\ell(N)$. In particular, is it true that $$\ell(N)\sim N^{1/2}?$$

Acceptance. FULLY RESOLVES (proof-shaped): a complete proof settling whether $\ell(N)\sim N^{1/2}$ — either proving $\ell(N)=(1+o(1))N^{1/2}$, or proving $\ell(N)\leq(c+o(1))N^{1/2}$ for some explicit $c<1$ together with a family of $N$-point sets attaining it — thereby fixing the asymptotic constant. A machine-checkable proof is preferred, otherwise a complete written proof. ADVANCES (each independently checkable): (a) construct $N$-point real sets whose largest Sidon subset has size $\leq(c+o(1))N^{1/2}$ with an explicit $c<1$, improving on the trivial $\{1,\ldots,N\}$ upper bound; (b) prove every $N$-set contains a Sidon subset of size $\geq(c'+o(1))N^{1/2}$ with an explicit $c'$ improving the Komlós–Sulyok–Szemerédi constant; (c) prove any nontrivial case of the Alon–Erdős partition conjecture; (d) certify exact values of $\ell(N)$ for a new range of small $N$ via a reproducible worst-case search with an exhaustiveness certificate. Deliver the proof/construction, or the search code plus certified values.