Erdős Problem #757
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
We say that `c` is admissible if, for any finite set `A` such that every four-element subset `B` determines at least five positive differences, there exists a Sidon subset `S` of size at least `c * A.ncard`. The difference condition is equivalent to `11 ≤ (B - B).ncard`.
Attempts
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Erdős asked for the best c such that every set of n reals in which any four points determine at least five distinct distances contains a Sidon set of size cn. The published bounds were 9/17 ≤ c ≤ 4/7 (Ma–Tang, February 2026). A reader's submission takes the 11-point set O = {0,49,75,83,98,101,117,119,121,159,269} and nests copies of it inside itself with φ(x) = 101 + 18x/269, which carries O's ends 0 and 269 onto two of its own interior points. We refereed it: for every depth K the set has 11 + 9K points, keeps the four-points-five-distances property, and its largest Sidon subset has exactly 7 + 5K points — so c ≤ 5/9. The finite facts are checked exhaustively (every 4-subset through 119 points, every additive relation through depth 12), the Sidon numbers by two independent solvers through 83 points, and a short 269-adic argument shows no relation can span more than ten levels, which turns the depth-12 census into a statement about every depth. One set suffices to beat 4/7: depth 6, 65 points, Sidon number 37. A hostile referee found four gaps in our first write-up, all fixable and fixed. The construction is the reader's; the interval is now 9/17 ≤ c ≤ 5/9.
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