For every integer d≥ 1 and every integer r≥ 2, let [r] denote a set of r vertices, and let K=[r][r]…[r] be the join of d+1 copies of [r]. For every continuous map f:K→R^d, there exist r pairwise vertex-disjoint rainbow d-simplices sigma_1,…,sigma_r of K whose vertices partition the vertices of K and such that f(sigma_1)cap…cap f(sigma_r)≠varnothing. Equivalently, each sigma_i contains exactly one vertex from each of the d+1 color classes, and the images of all r simplices have a common point.

Topological Bárány–Larman conjecture

For every integer $d\ge 1$ and every integer $r\ge 2$ , let $[r]$ denote a set of $r$ vertices, and let $K=[r]*[r]*\cdots*[r]$ be the join of $d+1$ copies of $[r]$ . For every continuous map $f:K\to\mathbb{R}^d$ , there exist $r$ pairwise vertex-disjoint rainbow $d$ -simplices $\sigma_1,\ldots,\sigma_r$ of $K$ whose vertices partition the vertices of $K$ and such that $f(\sigma_1)\cap\cdots\cap f(\sigma_r)\ne\varnothing$ . Equivalently, each $\sigma_i$ contains exactly one vertex from each of the $d+1$ color classes, and the images of all $r$ simplices have a common point.

References

Primary source

arXiv

Additional references

The topological Bárány-Larman conjecture for prime numbers — arXiv — Pablo Soberón

convex-geometry topological-combinatorics

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A September 2026 preprint claims the conjecture for prime numbers of parts and a failure of the proposed extension to prime powers, while the general case remains open.

The topological Bárány–Larman conjecture asserts that the sharp threshold for a colorful intersecting partition equals the number of parts, for every dimension and every number of parts. Bárány and Larman posed the geometric and topological versions in 1992.

Known results

Lovász proved the two-part case; the same argument gives the topological case for two parts.
Matschke and Ziegler (2009) proved the equality when one more than the number of parts is prime.
For general numbers of parts, the 2009 work gives bounds from the number of parts up to twice that number minus two, and an asymptotically sharp upper bound.

September 2026 claimed prime-case advance

A September 2026 preprint by Pablo Soberón claims a proof for prime numbers of parts and shows that the proposed optimal extension fails for prime powers. This is claimed progress, not an independently verified resolution; earlier literature states its prime-indexed result in the form “one more than the number of parts is prime.”

Current status (as of September 2026): The two-part case and the earlier one-less-than-a-prime cases are established; Soberón’s claimed prime-case advance and prime-power obstruction are unverified, and the conjecture for arbitrary numbers of parts remains open.

Sources

No solutions have been posted yet.