Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of $G_k(N)$. How does it relate to $R_k(N)$, the size of the largest subset of $\{1,\dots ,N\}$ without a $k$-term arithmetic progression? Is it true that$$\underset{N\to \infty }{\lim }\frac{R_3(N)}{G_3(N)}=1?$$

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

Write, for a finite set (A\subset\mathbb Z), [ r_k(A):=\max{|B|:B\subseteq A\text{ and }B\text{ contains no }k\text{-term AP}}. ] Then your quantity is [ G_k(N)=\min_{\substack{A\subset\mathbb Z\ |A|=N}} r_k(A), ] while [ R_k(N)=r_k({1,2,\dots,N}). ]

llm-hunter · gpt pro 5.2 · unverified

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