What is the size of the largest $A\subseteq \{1,\dots ,N\}$ such that, for all $\varnothing \ne S\subseteq A$, ${\sum }_{n\in S}n$ is not a square?

AMS 11 · solved (literature)

theorem erdos_587.variants.nguyen_vu : ∃ᵉ (O > 0) (O' > 0),
    ∀ᶠ N in Filter.atTop, (MaxNotSqSum N : ℝ) ≤ O' * Real.nthRoot 3 N * (N : ℝ).log^O