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  "body": "\n## Statement\n\nLet $A$ be a set of positive integers such that the series $\\sum_{a \\in A} 1/a$ diverges.\nThen for every integer $k \\ge 1$, the set $A$ contains an arithmetic progression of length $k$ with nonzero common difference.\n\n## Remarks\n\nThe conjecture is open for each $k \\ge 4$.\nThe case $k = 3$ was proved by Bloom and Sisask in 2020, using quantitative bounds in [[Roth's theorem|E7H8NWXESF]].\nThe set of primes satisfies the hypothesis, since the sum of the reciprocals of the primes diverges.\nSo the [[Green–Tao theorem|H8BAHWSEMC]] is a special case of the conjecture.\nEvery set of positive upper density satisfies the hypothesis, so the conjecture would strengthen [[Szemerédi's theorem|B23B0QEMN5]].\n\n## Explanations\n\n### L1 human\n\nThe conjecture says that a set of whole numbers that is not too sparse must contain evenly spaced rows of every length.\n\"Not too sparse\" means that the sum of the reciprocals of its elements is infinite.\nThe primes are an example, and for them the answer is known to be yes.\nFor general sets of this kind, the question is open beyond rows of three.\n",
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      "posed_by": "Paul Erdős"
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  "name": "Erdős conjecture on arithmetic progressions",
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