Let $F(k)$ be the minimal $N$ such that if we two-colour $\{1,\dots ,N\}$ there is a set $A$ of size $k$ such that all subset sums ${\sum }_{a\in S}a$ (for $\varnothing \ne S\subseteq A$) are monochromatic. Estimate $F(k)$.

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ \Sigma^*(A):=\\{\sum_{a\in S}a:\ \emptyset\neq S\subseteq A\\} ] (the nonempty subset–sums / finite sums set). Your $F(k)$ is exactly the **two–colour Folkman number for finite sums** (a finite version of Hindman/Folkman–Rado–Sanders). Its existence is guaranteed by **Folkman’s theorem**. ([Combinatorics][1])

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.