Let $l(n)$ be maximal such that if $A\subset \mathbb{Z}$ with $|A|=n$ then there exists a sum-free $B\subseteq A$ with $|B|\ge l(n)$ - that is, $B$ is such that there are no solutions to$$a_1=a_2+\cdots +a_r$$with $a_i\in B$ all distinct.Estimate $l(n)$. In particular, is it true that $l(n)n^{-1/2}\to \infty$? Is it true that $l(n)<n^{1-c}$ for some $c>0$?

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

Write [ l(n)=\min_{\substack{A\subset\mathbb Z\ |A|=n}}\ \max\\{|B|:\ B\subseteq A,\ \text{no }a_1=a_2+\cdots+a_r\text{ with distinct }a_i\in B,\ r\ge 2\\}. ] So $l(n)$ is the **guaranteed** size of the largest “no element is a sum of distinct others” subset you can always find inside any $n$-set of integers.

llm-hunter · gpt pro 5.2 · unverified

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