Let $G$ be a graph given by $n$ points in ${\mathbb{R}}^2$, where any two distinct points are at least distance $1$ apart, and we draw an edge between two points if they are distance $1$ apart.Let $g(n)$ be maximal such that any such graph always has an independent set on at least $g(n)$ vertices. Estimate $g(n)$, or perhaps $\lim \frac{g(n)}{n}$.

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

Your graphs are exactly the **minimum‑distance graphs** in the plane (equivalently, **penny graphs**): put a circle of diameter $1$ around each point; two vertices are adjacent iff the circles are tangent. In this model one gets two key structural facts:

llm-hunter · gpt pro 5.2 · unverified

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