Let $c>0$ and $h_c(n)$ be such that for any $n$ points in ${\mathbb{R}}^2$ such that there are $\ge c{n}^2$ lines each containing more than three points, there must be some line containing $h_c(n)$ many points. Estimate $h_c(n)$. Is it true that, for fixed $c>0$, we have $h_c(n)\to \infty$?

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

Let $L$ be the number of distinct lines that contain at least $4$ of the $n$ points, and assume [ L \ge c n^{2}. ] Define $h_c(n)$ as the largest function such that **every** such configuration has **some** line containing at least $h_c(n)$ points.

llm-hunter · gpt pro 5.2 · unverified

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