Breakthrough Tracker record
A claimed complete proof of the Erdős–Gyárfás monochromatic path-cover conjecture
The authors claim that, for every positive integer n, every red–blue edge-colouring of the complete graph K n has a same-colour collection of at most √n paths covering every vertex. This removes the finite-size gap left by the published large-n theorem.
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- Stable ID
math-erdos-gyarfas-monochromatic-path-covers-2026- Revision
math-erdos-gyarfas-monochromatic-path-covers-2026.v1- Field
- Mathematics · Extremal and Ramsey-style graph theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed proof
- Last checked
AI role
No AI role was disclosed in the inspected manuscript.
Record details
- Problem or result
- The 1995 Erdős–Gyárfás monochromatic path-cover conjecture
- Authors
- Hangdi Chen and Yaojun Chen
- Institutions
- Putian University; Nanjing University
- Result date
- Preprint submitted July 24, 2026
Why it matters
The conjecture asks for the sharp √n cover bound in every finite red–blue complete graph, a natural extremal question open since 1995. A 2026 Journal of Combinatorial Theory, Series B paper had proved it only for n greater than 20 40 .
Limits
This is a first-version preprint without located independent correctness assessment or peer review. The covering paths may intersect; the statement is not a vertex-disjoint path partition.
Sources
- Primary: Hangdi Chen and Yaojun Chen, On monochromatic path covers conjecture of Erdős–Gyárfás
- Published context: Pokrovskiy, Versteegen and Williams, published proof for all sufficiently large n
Correction and revision history
- 2026-07-27 — Added as a provisional claimed proof after full-text review; the finite-n scope, intersecting-path convention and lack of independent review are stated explicitly.
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