Is there a constant $c_t$, where $c_t\to \infty$ as $t\to \infty$, such that if $\mathcal{F}$ is a finite family of finite sets, all of size at least $t$, and for every set $X$ there are $<c_t|X|$ many $A\in \mathcal{F}$ with $A\subseteq X$, then $\mathcal{F}$ has chromatic number $2$ (in other words, has property B)?

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

No. In fact you can’t even have (c_t>2) for any $t$: for every $t$ there is a finite $(t+1)$-uniform family (\mathcal F) which is **not** 2‑colorable (has no property B) but still satisfies [ \forall X\qquad |\\{A\in\mathcal F:A\subseteq X\\}|\le 2|X|. ] So any admissible (c_t) would have to satisfy (c_t\le 2) for all …

llm-hunter · gpt pro 5.2 · unverified

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