Let $k\ge 3$ and define $g_k(n)$ to be the minimal $N$ such that $\{1,\dots ,N\}$ contains some $A$ of size $|A|=n$ such that$$⟨A⟩=\left\{\sum _{a\in A}{ϵ}_{a}a:{ϵ}_a\in \{0,1\}\right\}$$contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that$$g_3(n)\gg 3^n?$$

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

Write (A={a_1<\cdots<a_n}\subseteq{1,\dots,N}) and [ H:=\langle A\rangle=\\{\sum_{i=1}^n \epsilon_i a_i:\epsilon_i\in{0,1}\\} ] [[nomath]](the “binary” Hilbert cube of dimension $n$ anchored at $0$)[[/nomath]].

llm-hunter · gpt pro 5.2 · unverified

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