Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 48 min ago
PreprintCC BY 4.0arXiv

The Keevash--Mubayi simplex-cluster conjecture

  • Yongjiang Wu
  • Lihua Feng
Open at sourcePDFDOICite

Abstract

A d-simplex-cluster is a collection of d+1 distinct k-element sets with empty total intersection, nonempty intersection for every d members, and union of size at most 2k. We prove the simplex-cluster conjecture of Keevash and Mubayi, a common strengthening of the Erdős--Chvátal simplex conjecture and Mubayi's cluster conjecture. More precisely, for integers k>d≥2 and n≥ k(d+1)/d, every family of k-element subsets of an n-element set containing no d-simplex-cluster has at most binomn−1k−1 members. Equality holds if and only if the family consists of all k-element subsets containing a fixed point.

Related

  1. Complete counterexamples to Mubayi's supersaturation conjecture for graphs
    Liping Zhang, Qinghai Liu
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity
    Heng Li, Hong Liu, Xizhi Liu +1
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  3. The Erdős--Chvátal Simplex Conjecture
    Jing Huang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Regular simplex tensors: optimization landscape, conjecture proof, and beyond
    Lei Wang
    arXivPreprintCC BY-NC-SA 4.0Full textOptimisation & controlarXiv
  5. Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel
    Peigen Cao, Antoine de Saint Germain, Jiang-Hua Lu
    Forum of Mathematics, SigmaPeer-reviewedCC BY 4.01 citedGeometry & topologyCrossref

Cite

BibTeXRIS
BibTeX
@misc{wu2026keevashmubayi,
  title = {{The Keevash--Mubayi simplex-cluster conjecture}},
  author = {Yongjiang Wu and Lihua Feng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08567},
  eprint = {2610.08567},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08567},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Keevash--Mubayi simplex-cluster conjecture
AU  - Yongjiang Wu
AU  - Lihua Feng
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08567
UR  - https://arxiv.org/abs/2610.08567
AB  - A d-simplex-cluster is a collection of d+1 distinct k-element sets with empty total intersection, nonempty intersection for every d members, and union of size at most 2k. We prove the simplex-cluster conjecture of Keevash and Mubayi, a common strengthening of the Erdős--Chvátal simplex conjecture and Mubayi's cluster conjecture. More precisely, for integers k>d≥2 and n≥ k(d+1)/d, every family of k-element subsets of an n-element set containing no d-simplex-cluster has at most binomn−1k−1 members. Equality holds if and only if the family consists of all k-element subsets containing a fixed point.
ER  -