Let k>d≥ 2 and let n≥ k(d+1)/d. A d-simplex-cluster is a collection of d+1 distinct members A_1,…,A_d+1 of a family F⊆binom[n]k such that bigcap_i=1^d+1A_i=varnothing, bigcap_i≠ jA_i≠varnothing for every j∈[d+1], and |bigcup_i=1^d+1A_i|≤ 2k. The conjecture states that if F contains no d-simplex-cluster, then | F|≤binomn-1k-1. Moreover, equality holds if and only if there exists x∈[n] such that F=A∈binom[n]k:x∈ A, namely, F is a full star.

Keevash–Mubayi simplex-cluster conjecture

Let $k>d\ge 2$ and let $n\ge k(d+1)/d$ . A $d$ -simplex-cluster is a collection of $d+1$ distinct members $A_1,\dots,A_{d+1}$ of a family $\mathcal F\subseteq\binom{[n]}{k}$ such that $\bigcap_{i=1}^{d+1}A_i=\varnothing$ , $\bigcap_{i\ne j}A_i\ne\varnothing$ for every $j\in[d+1]$ , and $\left|\bigcup_{i=1}^{d+1}A_i\right|\le 2k$ . The conjecture states that if $\mathcal F$ contains no $d$ -simplex-cluster, then $|\mathcal F|\le\binom{n-1}{k-1}$ . Moreover, equality holds if and only if there exists $x\in[n]$ such that $\mathcal F=\{A\in\binom{[n]}{k}:x\in A\}$ , namely, $\mathcal F$ is a full star.

References

Primary source

arXiv

Additional references

The Keevash--Mubayi simplex-cluster conjecture — arXiv — Yongjiang Wu, Lihua Feng

extremal-combinatorics set-systems

Posted 11 hours ago · edited 11 hours ago

Progress summary

Refreshed 11 hours ago

Claimed solved

A new unrefereed paper claims the conjecture is completely proved, but the claim has not been independently checked.

Keevash and Mubayi’s conjecture predicts the star bound for every $k>d\ge2$ and $n\ge k(d+1)/d$ , with equality only for a full star. It is a strengthening of the Erdős–Chvátal simplex and Mubayi cluster conjectures.

Known results

Keevash–Mubayi: for sufficiently large $n$ in a specified linear range of $k$ , the star bound and equality case hold.
A 2018 paper: the bound holds for $d<k\le \frac{d}{d+1}n$ , with equality only for a star, plus an asymptotic version.
Currier, 2020: the conjecture is resolved for $4\le d+1\le k$ and $n\ge2k-d+2$ .

October 2026 claimed proof

Wu and Feng’s preprint claims that every cluster-free family satisfies $|\mathcal F|\le\binom{n-1}{k-1}$ for all $k>d\ge2$ and $n\ge k(d+1)/d$ , with equality only for a full star. This covers the entire conjectured range, but the retrieved record contains no independent verification or referee assessment.

Current status (as of October 2026): Earlier parameter ranges are proved, while the full conjecture remains open as a verified theorem because the claimed complete proof is unverified.

Sources

No solutions have been posted yet.