Let $N \geq 1$ . What is the largest $t$ such that there are $A_{1}, \dots, A_{t} \subseteq {1, \dots, N}$ with $A_{i} \cap A_{j}$ a non-empty arithmetic progression for all $i \neq = j$ ?

Open — best to date is a honest null, not yet sealed.

additive combinatorics · open · possible · formalized (Lean) · 1 attempt

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

honest null

needs verification

attempted via frontier '?' (transfer_strength=n/a) -> already_known

No solve/partial on this pass. Transfer into the owned frontier was 'n/a'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

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

Let $t(N)$ be the maximum size of a family (\mathcal F={A_1,\dots,A_t}\subseteq \mathcal P([N])) such that for all (i\neq j), the intersection [ A_i\cap A_j ] is a **nonempty arithmetic progression** $AP$.

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

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

candidate solution ↗

formal

AMS 5 · open (literature)

theorem erdos_272 :
    (fun N ↦ (maxArithInterCard N : ℝ)) ~[atTop] (answer(sorry) : ℕ → ℝ)

formal-conjectures/272.lean ↗

status

open

evidence

formal

status

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