1Context

Known frontier: Every planar thrackle satisfies m ≤ 1.393(n-1). The conjectured coefficient 1 is proved for straight-line thrackles and several other restricted drawing classes. Open boundary: Reduce the general coefficient to 1, or construct a planar thrackle with more edges than vertices. A recent higher-genus counterexample concerns a different conjecture and explicitly leaves the planar target open. A repository corpus search for thrackle returned no duplicate target.

2Problem setup

3What counts as a solution

2See also

• Constructing a union from non-cancelling intersectionscombinatorics
• Plane tilings by every five-cell lattice animalcombinatorics
• Cycle Double Cover Conjecturecombinatorics

“Conway’s thrackle conjecture.” TheoremDB. P3088. Problem statement; statement text SHA-256 c18b231fc33e70271a26407f8ec5916a791d98dd18a20a954af8bd9ad008e054. https://theoremdb.org/statement/?ref=P3088

@misc{theoremdb-problem-c18b231fc33e70271a26407f8ec5916a791d98dd18a20a954af8bd9ad008e054,
  title = {{Conway’s thrackle conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c18b231fc33e70271a26407f8ec5916a791d98dd18a20a954af8bd9ad008e054},
  url = {https://theoremdb.org/statement/?ref=P3088}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from doi.org^[1], current as of August 1, 2026.

1Lean verification

1References

1. Packet source. C. Hernández-Vélez, J. Kynčl, and G. Salazar, Thrackles on nonplanar surfaces, arXiv:2506.11808, version dated March 22, 2026. Abstract and Introduction, especially the current planar status paragraph. ↗ open copy ↗preprint · primary source · arXiv:2506.11808, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Explicitly states that Conway’s planar conjecture remains open and records m ≤ 1.393n as the best current general bound.Also cited at C. Hernández-Vélez, J. Kynčl, and G. Salazar, Thrackles on nonplanar surfaces, arXiv:2506.11808, version dated March 22, 2026. Abstract and Introduction, especially the current planar status paragraph.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: This is the dated publication status for the canonical target Conway’s thrackle conjecture.Source named by the research packet.
2. Yian Xu, “A New Upper Bound for Conway’s Thrackles”. Applied Mathematics and Computation 389 (2021), 125573. DOI 10.1016/j.amc.2020.125573. Abstract and main upper-bound theorem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves m ≤ 1.393(n-1) for simple connected thrackable graphs.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Proves m ≤ 1.393(n-1) for simple connected thrackable graphs.
3. Balázs Keszegh and Dániel Simon, “Convex hull thrackles”. Discrete Mathematics 349(3) (2026), 114840. DOI 10.1016/j.disc.2025.114840. Abstract and opening discussion of linear thrackles. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Records the planar conjecture and the known exact result for straight-line thrackles while studying a convex-hull extension.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Records the planar conjecture and the known exact result for straight-line thrackles while studying a convex-hull extension.
4. Radoslav Fulek and János Pach, “A computational approach to Conwayʼs thrackle conjecture”. Computational Geometry 44(6-7) (2011), 345-355. DOI 10.1016/j.comgeo.2011.02.001. Abstract and main algorithmic upper-bound result. ↗ open copy ↗journal article · primary source · checked 2026-08-01Source use: original summary.Gives a finite procedure for testing uniform bounds of the form t(n)<(1+ε)n and established the earlier 167n/117 bound.Source used to assess the problem's recorded status.For Conway’s thrackle conjecture: Gives a finite procedure for testing uniform bounds of the form t(n)<(1+ε)n and established the earlier 167n/117 bound.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.