The board / graph theory / #1007
Erdős Problem #1007
SOLVED LEANdatabase updated 2026-01-19
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The complete tripartite graph \(K_{1,3,3}\).
The dimension of a graph \(G\) is the minimal \(n\) such that \(G\) can be embedded in \(\mathbb{R}^n\) such that every edge of \(G\) is a unit line segment. What is the smallest number of edges in a graph with dimension \(4\)? Answer: The smallest number of edges is \(9\), achieved solely by \(K_{3,3}\), proved by House [Ho13]. An alternative proof was given by Chaffee and Noble [ChNo16], who also prove that the smallest number of edges in a graph of dimension \(5\) is \(15\) (achieved by \(K_6\) and \(K_{1,3,3}\)).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1007.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1007
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.