A pairwise balanced design for ${1, \dots, n}$ is a collection of sets $A_{1}, \dots, A_{m} \subseteq {1, \dots, n}$ such that $2 \leq ∣ A_{i} ∣ < n$ and every pair of distinct elements $x, y \in {1, \dots, n}$ is contained in exactly one $A_{i}$ .Is there a constant $C > 0$ and, for all large $n$ , a pairwise balanced design such that $∣ A_{i} ∣ > n^{1/2} - C$ for all $1 \leq i \leq m$ ?

Worked, still open.

combinatorics · open · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

unverified AI candidates (2)

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

Yes — as stated, the problem is *trivial* because you are allowed to take $m=1$.

llm-hunter · gpt pro 5.2 · unverified

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

open