Let $F_k(n)$ be minimal such that for any $n$ points in ${\mathbb{R}}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ of the points, and $f_k(n)$ similarly but with lines passing through exactly $k$ points.Estimate $f_k(n)$ and $F_k(n)$ - in particular, determine $\lim F_k(n)/n^2$ and $\lim f_k(n)/n^2$.

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

Let (P\subset \mathbb R^2) with (|P|=n). Write [ t_j(P):=|\\{\text{lines }\ell:\ |\ell\cap P|=j\\}|,\qquad T_k(P):=|\\{\text{lines }\ell:\ |\ell\cap P|\ge k\\}|=\sum_{j\ge k} t_j(P). ] Then your extremal functions are [ f_k(n)=\max_{|P|=n} t_k(P),\qquad F_k(n)=\max_{|P|=n} T_k(P)=\max_{|P|=n}\sum_{j\ge k} t_j(P). ]

llm-hunter · gpt pro 5.2 · unverified

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