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OP-12002OpenNumber theory

Sidon set density inside (4,5) sets (C5bC_{5b})

Sidon set density inside (4,5) sets: determine C5bC_{5b} or improve its bounds; the repository records 0.5294 (lower) and 0.5714 (upper).

MSC 11B83 · Special sequences and polynomials

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C5bC_{5b} is the largest constant such that every (4,5)(4,5)-set of size nn (i.e., a set of reals such that every four-element subset determines at least five distinct differences) contains a Sidon set of cardinality C5bnC_{5b}n.

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Background

References (from the repository page)

  • [GL95] A. Gyárfás and J. Lehel, Linear sets with five distinct differences among any four elements, J. Combin. Theory Ser. B 64 (1995), 108–118.
  • [MT26] Jie Ma and Quanyu Tang, Largest Sidon subsets in weak Sidon sets. arXiv:2602.23282 (2026).

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.5294; best known upper bound 0.5714. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11 Trivial
35=0.6\frac{3}{5} = 0.6 [GL95]
47≈0.5714\frac{4}{7} \approx 0.5714 [MT26]

Known lower bounds

Bound Reference Comments
12\frac{1}{2} [GL95] A short 2-colorability argument
12+1141×76≈0.500093\frac{1}{2} + \frac{1}{141 \times 76} \approx 0.500093 [GL95]
917≈0.5294\frac{9}{17} \approx 0.5294 [MT26]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Sidon set density inside (4,5) sets ($C_{5b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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