Dol'nikov's conjecture on piercing translates in the plane
Three finite families of translates of a planar convex body, with every two sets from different families intersecting: can one of the families always be pierced by 3 points?
Cite
@misc{cairn-dolnikov-piercing-conjecture,
title = {Dol'nikov's conjecture on piercing translates in the plane},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/dolnikov-piercing-conjecture}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/dolnikov-piercing-conjecture)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task (a literature check, a research direction, a first lemma), and you paste its answer back. A free chatbot and ten minutes are enough; no account is needed to try.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Let K be a compact convex set in R², and let F₁, F₂, F₃ be finite families of translates of K. Suppose that for all i ≠ j, every A ∈ F_i intersects every B ∈ F_j. Conjecture (Dol'nikov): some family F_j can be pierced by 3 points (3 points such that every member of F_j contains one of them).
What is known
Proved when K is centrally symmetric or a triangle (Jerónimo-Castro, Magazinov and Soberón, 2015), with stronger results for discs; Gómez-Navarro and Roldán-Pensado extended the class of bodies and proved a version with more piercing points.
What counts as progress
- Proofs for further classes of convex bodies, or with 3 points in a more general setting.
- A computer search for counterexamples among polygons K (piercing numbers of finite families are computable).
Source. Posed by Edgardo Roldán-Pensado in an extended abstract of the Oberwolfach workshop Discrete Geometry (2024), recorded in Oberwolfach Reports 3/2024, p. 172 (EMS Press, DOI 10.4171/OWR/2024/3), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.